%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Universal_Hash_Families/0033_Preliminary_Results/prob_00094_003582__18489984_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1607 ( 426 unt; 329 typ;   0 def)
%            Number of atoms       : 3968 ( 919 equ;   0 cnn)
%            Maximal formula atoms :   10 (   3 avg)
%            Number of connectives : 15537 (  84   ~;  26   |; 444   &;13277   @)
%                                         (   0 <=>;1706  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   8 avg)
%            Number of types       :   52 (  51 usr)
%            Number of type conns  : 1994 (1994   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  281 ( 278 usr;   6 con; 0-3 aty)
%            Number of variables   : 4786 ( 918   ^;3819   !;  49   ?;4786   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:44:47.841
%------------------------------------------------------------------------------
% Could-be-implicit typings (51)
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__d_J_Mt__Set__Oset_Itf__d_J_J_J,type,
    set_set_d_set_d: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__d_J_Mt__Set__Oset_Itf__b_J_J_J,type,
    set_set_d_set_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__b_J_Mt__Set__Oset_Itf__b_J_J_J,type,
    set_set_b_set_b: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__d_J_J,type,
    sigma_measure_set_d: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__b_J_J,type,
    sigma_measure_set_b: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__d_Mtf__d_J_J,type,
    sigma_measure_d_d: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__d_Mtf__b_J_J,type,
    sigma_measure_d_b: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__b_Mtf__d_J_J,type,
    sigma_measure_b_d: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__b_Mtf__b_J_J,type,
    sigma_measure_b_b: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__d_J_J_J,type,
    set_set_set_d: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
    set_set_set_b: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_Itf__d_Mtf__d_J_J_J,type,
    set_set_d_d: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_Itf__d_Mtf__b_J_J_J,type,
    set_set_d_b: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_Itf__b_Mtf__d_J_J_J,type,
    set_set_b_d: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_Itf__b_Mtf__b_J_J_J,type,
    set_set_b_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__d_Mt__Set__Oset_Itf__b_J_J_J,type,
    set_d_set_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__c_Mt__Set__Oset_Itf__d_J_J_J,type,
    set_c_set_d: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__c_Mt__Set__Oset_Itf__b_J_J_J,type,
    set_c_set_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_Mt__Set__Oset_Itf__b_J_J_J,type,
    set_b_set_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__d_J_Mtf__c_J_J,type,
    set_set_d_c: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__d_J_Mtf__b_J_J,type,
    set_set_d_b2: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__b_J_Mtf__d_J_J,type,
    set_set_b_d2: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__b_J_Mtf__c_J_J,type,
    set_set_b_c: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__b_J_Mtf__b_J_J,type,
    set_set_b_b2: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_Itf__d_Mtf__d_J_Mtf__b_J_J,type,
    set_d_d_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_Itf__d_Mtf__b_J_Mtf__b_J_J,type,
    set_d_b_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_Itf__b_Mtf__d_J_Mtf__b_J_J,type,
    set_b_d_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_Itf__b_Mtf__b_J_Mtf__b_J_J,type,
    set_b_b_b: $tType ).

thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
    extend8495563244428889912nnreal: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__d_J,type,
    sigma_measure_d: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__c_J,type,
    sigma_measure_c: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    sigma_measure_b: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__d_J_J,type,
    set_set_d: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
    set_set_c: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    set_set_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__d_Mtf__d_J_J,type,
    set_d_d: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__d_Mtf__c_J_J,type,
    set_d_c: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__d_Mtf__b_J_J,type,
    set_d_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__d_J_J,type,
    set_c_d: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__c_J_J,type,
    set_c_c: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__b_J_J,type,
    set_c_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__d_J_J,type,
    set_b_d: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__c_J_J,type,
    set_b_c: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
    set_b_b: $tType ).

thf(ty_n_t__Set__Oset_Itf__d_J,type,
    set_d: $tType ).

thf(ty_n_t__Set__Oset_Itf__c_J,type,
    set_c: $tType ).

thf(ty_n_t__Set__Oset_Itf__b_J,type,
    set_b: $tType ).

thf(ty_n_t__Real__Oreal,type,
    real: $tType ).

thf(ty_n_tf__d,type,
    d: $tType ).

thf(ty_n_tf__c,type,
    c: $tType ).

thf(ty_n_tf__b,type,
    b: $tType ).

% Explicit typings (278)
thf(sy_c_Complete__Measure_Onull__part_001tf__b,type,
    complete_null_part_b: sigma_measure_b > set_b > set_b ).

thf(sy_c_Complete__Measure_Onull__part_001tf__d,type,
    complete_null_part_d: sigma_measure_d > set_d > set_d ).

thf(sy_c_Complete__Measure_Oouter__measure__of_001t__Set__Oset_Itf__b_J,type,
    comple3418655172853872587_set_b: sigma_measure_set_b > set_set_b > extend8495563244428889912nnreal ).

thf(sy_c_Complete__Measure_Oouter__measure__of_001tf__b,type,
    comple3326625653960060651e_of_b: sigma_measure_b > set_b > extend8495563244428889912nnreal ).

thf(sy_c_Complete__Measure_Oouter__measure__of_001tf__d,type,
    comple3326625653960060653e_of_d: sigma_measure_d > set_d > extend8495563244428889912nnreal ).

thf(sy_c_FuncSet_OPi_001_062_Itf__b_Mtf__b_J_001tf__b,type,
    pi_b_b_b: set_b_b > ( ( b > b ) > set_b ) > set_b_b_b ).

thf(sy_c_FuncSet_OPi_001_062_Itf__b_Mtf__d_J_001tf__b,type,
    pi_b_d_b: set_b_d > ( ( b > d ) > set_b ) > set_b_d_b ).

thf(sy_c_FuncSet_OPi_001_062_Itf__d_Mtf__b_J_001tf__b,type,
    pi_d_b_b: set_d_b > ( ( d > b ) > set_b ) > set_d_b_b ).

thf(sy_c_FuncSet_OPi_001_062_Itf__d_Mtf__d_J_001tf__b,type,
    pi_d_d_b: set_d_d > ( ( d > d ) > set_b ) > set_d_d_b ).

thf(sy_c_FuncSet_OPi_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
    pi_set_b_set_b: set_set_b > ( set_b > set_set_b ) > set_set_b_set_b ).

thf(sy_c_FuncSet_OPi_001t__Set__Oset_Itf__b_J_001tf__b,type,
    pi_set_b_b: set_set_b > ( set_b > set_b ) > set_set_b_b2 ).

thf(sy_c_FuncSet_OPi_001t__Set__Oset_Itf__b_J_001tf__c,type,
    pi_set_b_c: set_set_b > ( set_b > set_c ) > set_set_b_c ).

thf(sy_c_FuncSet_OPi_001t__Set__Oset_Itf__b_J_001tf__d,type,
    pi_set_b_d: set_set_b > ( set_b > set_d ) > set_set_b_d2 ).

thf(sy_c_FuncSet_OPi_001t__Set__Oset_Itf__d_J_001t__Set__Oset_Itf__b_J,type,
    pi_set_d_set_b: set_set_d > ( set_d > set_set_b ) > set_set_d_set_b ).

thf(sy_c_FuncSet_OPi_001t__Set__Oset_Itf__d_J_001t__Set__Oset_Itf__d_J,type,
    pi_set_d_set_d: set_set_d > ( set_d > set_set_d ) > set_set_d_set_d ).

thf(sy_c_FuncSet_OPi_001t__Set__Oset_Itf__d_J_001tf__b,type,
    pi_set_d_b: set_set_d > ( set_d > set_b ) > set_set_d_b2 ).

thf(sy_c_FuncSet_OPi_001t__Set__Oset_Itf__d_J_001tf__c,type,
    pi_set_d_c: set_set_d > ( set_d > set_c ) > set_set_d_c ).

thf(sy_c_FuncSet_OPi_001tf__b_001t__Set__Oset_Itf__b_J,type,
    pi_b_set_b: set_b > ( b > set_set_b ) > set_b_set_b ).

thf(sy_c_FuncSet_OPi_001tf__b_001tf__b,type,
    pi_b_b: set_b > ( b > set_b ) > set_b_b ).

thf(sy_c_FuncSet_OPi_001tf__b_001tf__d,type,
    pi_b_d: set_b > ( b > set_d ) > set_b_d ).

thf(sy_c_FuncSet_OPi_001tf__c_001t__Set__Oset_Itf__b_J,type,
    pi_c_set_b: set_c > ( c > set_set_b ) > set_c_set_b ).

thf(sy_c_FuncSet_OPi_001tf__c_001t__Set__Oset_Itf__d_J,type,
    pi_c_set_d: set_c > ( c > set_set_d ) > set_c_set_d ).

thf(sy_c_FuncSet_OPi_001tf__c_001tf__b,type,
    pi_c_b: set_c > ( c > set_b ) > set_c_b ).

thf(sy_c_FuncSet_OPi_001tf__c_001tf__c,type,
    pi_c_c: set_c > ( c > set_c ) > set_c_c ).

thf(sy_c_FuncSet_OPi_001tf__d_001t__Set__Oset_Itf__b_J,type,
    pi_d_set_b: set_d > ( d > set_set_b ) > set_d_set_b ).

thf(sy_c_FuncSet_OPi_001tf__d_001tf__b,type,
    pi_d_b: set_d > ( d > set_b ) > set_d_b ).

thf(sy_c_FuncSet_OPi_001tf__d_001tf__d,type,
    pi_d_d: set_d > ( d > set_d ) > set_d_d ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
    zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).

thf(sy_c_If_001t__Set__Oset_Itf__d_J,type,
    if_set_d: $o > set_d > set_d > set_d ).

thf(sy_c_If_001tf__b,type,
    if_b: $o > b > b > b ).

thf(sy_c_If_001tf__c,type,
    if_c: $o > c > c > c ).

thf(sy_c_If_001tf__d,type,
    if_d: $o > d > d > d ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_Itf__b_Mtf__b_J_M_Eo_J,type,
    inf_inf_b_b_o: ( ( b > b ) > $o ) > ( ( b > b ) > $o ) > ( b > b ) > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_Itf__b_Mtf__d_J_M_Eo_J,type,
    inf_inf_b_d_o: ( ( b > d ) > $o ) > ( ( b > d ) > $o ) > ( b > d ) > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_Itf__d_Mtf__b_J_M_Eo_J,type,
    inf_inf_d_b_o: ( ( d > b ) > $o ) > ( ( d > b ) > $o ) > ( d > b ) > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_Itf__d_Mtf__d_J_M_Eo_J,type,
    inf_inf_d_d_o: ( ( d > d ) > $o ) > ( ( d > d ) > $o ) > ( d > d ) > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_Itf__b_J_M_Eo_J,type,
    inf_inf_set_b_o: ( set_b > $o ) > ( set_b > $o ) > set_b > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_Itf__d_J_M_Eo_J,type,
    inf_inf_set_d_o: ( set_d > $o ) > ( set_d > $o ) > set_d > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__b_M_Eo_J,type,
    inf_inf_b_o: ( b > $o ) > ( b > $o ) > b > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__c_M_Eo_J,type,
    inf_inf_c_o: ( c > $o ) > ( c > $o ) > c > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__d_M_Eo_J,type,
    inf_inf_d_o: ( d > $o ) > ( d > $o ) > d > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
    inf_inf_set_b_b: set_b_b > set_b_b > set_b_b ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__b_Mtf__d_J_J,type,
    inf_inf_set_b_d: set_b_d > set_b_d > set_b_d ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__d_Mtf__b_J_J,type,
    inf_inf_set_d_b: set_d_b > set_d_b > set_d_b ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__d_Mtf__d_J_J,type,
    inf_inf_set_d_d: set_d_d > set_d_d > set_d_d ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
    inf_in7907464295584518029_set_b: set_set_set_b > set_set_set_b > set_set_set_b ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    inf_inf_set_set_b: set_set_b > set_set_b > set_set_b ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__d_J_J,type,
    inf_inf_set_set_d: set_set_d > set_set_d > set_set_d ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
    inf_inf_set_b: set_b > set_b > set_b ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__c_J,type,
    inf_inf_set_c: set_c > set_c > set_c ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__d_J,type,
    inf_inf_set_d: set_d > set_d > set_d ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
    sup_sup_set_b_b: set_b_b > set_b_b > set_b_b ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__b_Mtf__d_J_J,type,
    sup_sup_set_b_d: set_b_d > set_b_d > set_b_d ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__d_Mtf__b_J_J,type,
    sup_sup_set_d_b: set_d_b > set_d_b > set_d_b ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__d_Mtf__d_J_J,type,
    sup_sup_set_d_d: set_d_d > set_d_d > set_d_d ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    sup_sup_set_set_b: set_set_b > set_set_b > set_set_b ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__d_J_J,type,
    sup_sup_set_set_d: set_set_d > set_set_d > set_set_d ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
    sup_sup_set_b: set_b > set_b > set_b ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__c_J,type,
    sup_sup_set_c: set_c > set_c > set_c ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__d_J,type,
    sup_sup_set_d: set_d > set_d > set_d ).

thf(sy_c_Measure__Space_Odistr_001t__Set__Oset_Itf__b_J_001tf__b,type,
    measur6428830690898164184et_b_b: sigma_measure_set_b > sigma_measure_b > ( set_b > b ) > sigma_measure_b ).

thf(sy_c_Measure__Space_Odistr_001t__Set__Oset_Itf__b_J_001tf__d,type,
    measur6428830690898164186et_b_d: sigma_measure_set_b > sigma_measure_d > ( set_b > d ) > sigma_measure_d ).

thf(sy_c_Measure__Space_Odistr_001t__Set__Oset_Itf__d_J_001tf__b,type,
    measur5157532689688190806et_d_b: sigma_measure_set_d > sigma_measure_b > ( set_d > b ) > sigma_measure_b ).

thf(sy_c_Measure__Space_Odistr_001t__Set__Oset_Itf__d_J_001tf__d,type,
    measur5157532689688190808et_d_d: sigma_measure_set_d > sigma_measure_d > ( set_d > d ) > sigma_measure_d ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001t__Set__Oset_Itf__b_J,type,
    measur8192432674156524440_set_b: sigma_measure_b > sigma_measure_set_b > ( b > set_b ) > sigma_measure_set_b ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001tf__b,type,
    measure_distr_b_b: sigma_measure_b > sigma_measure_b > ( b > b ) > sigma_measure_b ).

thf(sy_c_Measure__Space_Odistr_001tf__b_001tf__d,type,
    measure_distr_b_d: sigma_measure_b > sigma_measure_d > ( b > d ) > sigma_measure_d ).

thf(sy_c_Measure__Space_Odistr_001tf__c_001tf__b,type,
    measure_distr_c_b: sigma_measure_c > sigma_measure_b > ( c > b ) > sigma_measure_b ).

thf(sy_c_Measure__Space_Odistr_001tf__c_001tf__d,type,
    measure_distr_c_d: sigma_measure_c > sigma_measure_d > ( c > d ) > sigma_measure_d ).

thf(sy_c_Measure__Space_Odistr_001tf__d_001t__Set__Oset_Itf__b_J,type,
    measur1657990710920724502_set_b: sigma_measure_d > sigma_measure_set_b > ( d > set_b ) > sigma_measure_set_b ).

thf(sy_c_Measure__Space_Odistr_001tf__d_001tf__b,type,
    measure_distr_d_b: sigma_measure_d > sigma_measure_b > ( d > b ) > sigma_measure_b ).

thf(sy_c_Measure__Space_Odistr_001tf__d_001tf__d,type,
    measure_distr_d_d: sigma_measure_d > sigma_measure_d > ( d > d ) > sigma_measure_d ).

thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__b_J_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    measur8864879078030831187_set_b: set_set_set_b > ( set_set_b > set_set_b ) > $o ).

thf(sy_c_Measure__Space_Oincreasing_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
    measur6668990532724476275_set_b: set_set_set_b > ( set_set_b > set_b ) > $o ).

thf(sy_c_Measure__Space_Oincreasing_001tf__b_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    measur1967322967037640499_set_b: set_set_b > ( set_b > set_set_b ) > $o ).

thf(sy_c_Measure__Space_Oincreasing_001tf__b_001t__Set__Oset_Itf__b_J,type,
    measur9187034394192042579_set_b: set_set_b > ( set_b > set_b ) > $o ).

thf(sy_c_Measure__Space_Oincreasing_001tf__d_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    measur7802506524085621681_set_b: set_set_d > ( set_d > set_set_b ) > $o ).

thf(sy_c_Measure__Space_Oincreasing_001tf__d_001t__Set__Oset_Itf__b_J,type,
    measur2652592430956242641_set_b: set_set_d > ( set_d > set_b ) > $o ).

thf(sy_c_Measure__Space_Onull__sets_001_062_Itf__b_Mtf__b_J,type,
    measur1875269523382374962ts_b_b: sigma_measure_b_b > set_set_b_b ).

thf(sy_c_Measure__Space_Onull__sets_001_062_Itf__b_Mtf__d_J,type,
    measur1875269531988832564ts_b_d: sigma_measure_b_d > set_set_b_d ).

thf(sy_c_Measure__Space_Onull__sets_001_062_Itf__d_Mtf__b_J,type,
    measur5523730396291798704ts_d_b: sigma_measure_d_b > set_set_d_b ).

thf(sy_c_Measure__Space_Onull__sets_001_062_Itf__d_Mtf__d_J,type,
    measur5523730404898256306ts_d_d: sigma_measure_d_d > set_set_d_d ).

thf(sy_c_Measure__Space_Onull__sets_001t__Set__Oset_Itf__b_J,type,
    measur1516554132335629586_set_b: sigma_measure_set_b > set_set_set_b ).

thf(sy_c_Measure__Space_Onull__sets_001t__Set__Oset_Itf__d_J,type,
    measur1516554140942087188_set_d: sigma_measure_set_d > set_set_set_d ).

thf(sy_c_Measure__Space_Onull__sets_001tf__b,type,
    measure_null_sets_b: sigma_measure_b > set_set_b ).

thf(sy_c_Measure__Space_Onull__sets_001tf__c,type,
    measure_null_sets_c: sigma_measure_c > set_set_c ).

thf(sy_c_Measure__Space_Onull__sets_001tf__d,type,
    measure_null_sets_d: sigma_measure_d > set_set_d ).

thf(sy_c_Measure__Space_Osup__measure_H_001t__Set__Oset_Itf__b_J,type,
    measur132476219272276657_set_b: sigma_measure_set_b > sigma_measure_set_b > sigma_measure_set_b ).

thf(sy_c_Measure__Space_Osup__measure_H_001tf__b,type,
    measur3004909623614618065sure_b: sigma_measure_b > sigma_measure_b > sigma_measure_b ).

thf(sy_c_Measure__Space_Osup__measure_H_001tf__d,type,
    measur3004909623614618067sure_d: sigma_measure_d > sigma_measure_d > sigma_measure_d ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__b_Mtf__b_J_M_Eo_J,type,
    ord_less_eq_b_b_o: ( ( b > b ) > $o ) > ( ( b > b ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__b_Mtf__d_J_M_Eo_J,type,
    ord_less_eq_b_d_o: ( ( b > d ) > $o ) > ( ( b > d ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__d_Mtf__b_J_M_Eo_J,type,
    ord_less_eq_d_b_o: ( ( d > b ) > $o ) > ( ( d > b ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__d_Mtf__d_J_M_Eo_J,type,
    ord_less_eq_d_d_o: ( ( d > d ) > $o ) > ( ( d > d ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Set__Oset_Itf__b_J_J_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    ord_le3779468636425607816nnreal: ( set_set_b > extend8495563244428889912nnreal ) > ( set_set_b > extend8495563244428889912nnreal ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__b_J_M_Eo_J,type,
    ord_less_eq_set_b_o: ( set_b > $o ) > ( set_b > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__b_J_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    ord_le6365330890766322088nnreal: ( set_b > extend8495563244428889912nnreal ) > ( set_b > extend8495563244428889912nnreal ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__d_J_M_Eo_J,type,
    ord_less_eq_set_d_o: ( set_d > $o ) > ( set_d > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__d_J_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    ord_le5694847263963582634nnreal: ( set_d > extend8495563244428889912nnreal ) > ( set_d > extend8495563244428889912nnreal ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__b_M_Eo_J,type,
    ord_less_eq_b_o: ( b > $o ) > ( b > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__c_M_Eo_J,type,
    ord_less_eq_c_o: ( c > $o ) > ( c > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
    ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_Itf__b_Mtf__b_J_Mtf__b_J_J,type,
    ord_le7124591221154113873_b_b_b: set_b_b_b > set_b_b_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_Itf__b_Mtf__d_J_Mtf__b_J_J,type,
    ord_le3202560045296529359_b_d_b: set_b_d_b > set_b_d_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_Itf__d_Mtf__b_J_Mtf__b_J_J,type,
    ord_le1567182605434068819_d_b_b: set_d_b_b > set_d_b_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_Itf__d_Mtf__d_J_Mtf__b_J_J,type,
    ord_le6868523466431260113_d_d_b: set_d_d_b > set_d_d_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Set__Oset_Itf__b_J_Mt__Set__Oset_Itf__b_J_J_J,type,
    ord_le1406319396829752251_set_b: set_set_b_set_b > set_set_b_set_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Set__Oset_Itf__b_J_Mtf__b_J_J,type,
    ord_le7727052776605854427et_b_b: set_set_b_b2 > set_set_b_b2 > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Set__Oset_Itf__d_J_Mt__Set__Oset_Itf__b_J_J_J,type,
    ord_le84730631212153401_set_b: set_set_d_set_b > set_set_d_set_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Set__Oset_Itf__d_J_Mtf__b_J_J,type,
    ord_le3805021600748269913et_d_b: set_set_d_b2 > set_set_d_b2 > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
    ord_less_eq_set_b_b: set_b_b > set_b_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__b_Mtf__d_J_J,type,
    ord_less_eq_set_b_d: set_b_d > set_b_d > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__c_Mt__Set__Oset_Itf__b_J_J_J,type,
    ord_le213790835969874778_set_b: set_c_set_b > set_c_set_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__c_Mtf__b_J_J,type,
    ord_less_eq_set_c_b: set_c_b > set_c_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__d_Mtf__b_J_J,type,
    ord_less_eq_set_d_b: set_d_b > set_d_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__d_Mtf__d_J_J,type,
    ord_less_eq_set_d_d: set_d_d > set_d_d > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
    ord_le3201067847557142847_set_b: set_set_set_b > set_set_set_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    ord_le3795704787696855135_set_b: set_set_b > set_set_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__d_J_J,type,
    ord_le3937772867789547105_set_d: set_set_d > set_set_d > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
    ord_less_eq_set_b: set_b > set_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__c_J,type,
    ord_less_eq_set_c: set_c > set_c > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__d_J,type,
    ord_less_eq_set_d: set_d > set_d > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__b_J_J,type,
    ord_le5713619651007674940_set_b: sigma_measure_set_b > sigma_measure_set_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    ord_le254669799889008988sure_b: sigma_measure_b > sigma_measure_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_Itf__d_J,type,
    ord_le254669808495466590sure_d: sigma_measure_d > sigma_measure_d > $o ).

thf(sy_c_Set_OBall_001_062_Itf__b_Mtf__b_J,type,
    ball_b_b: set_b_b > ( ( b > b ) > $o ) > $o ).

thf(sy_c_Set_OBall_001_062_Itf__b_Mtf__d_J,type,
    ball_b_d: set_b_d > ( ( b > d ) > $o ) > $o ).

thf(sy_c_Set_OBall_001_062_Itf__d_Mtf__b_J,type,
    ball_d_b: set_d_b > ( ( d > b ) > $o ) > $o ).

thf(sy_c_Set_OBall_001_062_Itf__d_Mtf__d_J,type,
    ball_d_d: set_d_d > ( ( d > d ) > $o ) > $o ).

thf(sy_c_Set_OBall_001t__Set__Oset_Itf__b_J,type,
    ball_set_b: set_set_b > ( set_b > $o ) > $o ).

thf(sy_c_Set_OBall_001t__Set__Oset_Itf__d_J,type,
    ball_set_d: set_set_d > ( set_d > $o ) > $o ).

thf(sy_c_Set_OBall_001tf__b,type,
    ball_b: set_b > ( b > $o ) > $o ).

thf(sy_c_Set_OBall_001tf__c,type,
    ball_c: set_c > ( c > $o ) > $o ).

thf(sy_c_Set_OCollect_001_062_It__Set__Oset_Itf__b_J_Mtf__b_J,type,
    collect_set_b_b: ( ( set_b > b ) > $o ) > set_set_b_b2 ).

thf(sy_c_Set_OCollect_001_062_It__Set__Oset_Itf__b_J_Mtf__d_J,type,
    collect_set_b_d: ( ( set_b > d ) > $o ) > set_set_b_d2 ).

thf(sy_c_Set_OCollect_001_062_Itf__b_Mtf__b_J,type,
    collect_b_b: ( ( b > b ) > $o ) > set_b_b ).

thf(sy_c_Set_OCollect_001_062_Itf__b_Mtf__d_J,type,
    collect_b_d: ( ( b > d ) > $o ) > set_b_d ).

thf(sy_c_Set_OCollect_001_062_Itf__d_Mtf__b_J,type,
    collect_d_b: ( ( d > b ) > $o ) > set_d_b ).

thf(sy_c_Set_OCollect_001_062_Itf__d_Mtf__d_J,type,
    collect_d_d: ( ( d > d ) > $o ) > set_d_d ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    collect_set_set_b: ( set_set_b > $o ) > set_set_set_b ).

thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
    collect_set_b: ( set_b > $o ) > set_set_b ).

thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__d_J,type,
    collect_set_d: ( set_d > $o ) > set_set_d ).

thf(sy_c_Set_OCollect_001tf__b,type,
    collect_b: ( b > $o ) > set_b ).

thf(sy_c_Set_OCollect_001tf__c,type,
    collect_c: ( c > $o ) > set_c ).

thf(sy_c_Set_OCollect_001tf__d,type,
    collect_d: ( d > $o ) > set_d ).

thf(sy_c_Set_OPow_001t__Set__Oset_Itf__b_J,type,
    pow_set_b: set_set_b > set_set_set_b ).

thf(sy_c_Set_OPow_001tf__b,type,
    pow_b: set_b > set_set_b ).

thf(sy_c_Set_OPow_001tf__d,type,
    pow_d: set_d > set_set_d ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__b_J_001_062_Itf__b_Mtf__b_J,type,
    vimage_set_b_b_b: ( set_b > b > b ) > set_b_b > set_set_b ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__b_J_001_062_Itf__b_Mtf__d_J,type,
    vimage_set_b_b_d: ( set_b > b > d ) > set_b_d > set_set_b ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__b_J_001_062_Itf__d_Mtf__b_J,type,
    vimage_set_b_d_b: ( set_b > d > b ) > set_d_b > set_set_b ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
    vimage_set_b_set_b: ( set_b > set_b ) > set_set_b > set_set_b ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__d_J,type,
    vimage_set_b_set_d: ( set_b > set_d ) > set_set_d > set_set_b ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__b_J_001tf__b,type,
    vimage_set_b_b: ( set_b > b ) > set_b > set_set_b ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__b_J_001tf__c,type,
    vimage_set_b_c: ( set_b > c ) > set_c > set_set_b ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__b_J_001tf__d,type,
    vimage_set_b_d: ( set_b > d ) > set_d > set_set_b ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__d_J_001t__Set__Oset_Itf__d_J,type,
    vimage_set_d_set_d: ( set_d > set_d ) > set_set_d > set_set_d ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_Itf__d_J_001tf__c,type,
    vimage_set_d_c: ( set_d > c ) > set_c > set_set_d ).

thf(sy_c_Set_Ovimage_001tf__b_001t__Set__Oset_Itf__b_J,type,
    vimage_b_set_b: ( b > set_b ) > set_set_b > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001t__Set__Oset_Itf__d_J,type,
    vimage_b_set_d: ( b > set_d ) > set_set_d > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001tf__b,type,
    vimage_b_b: ( b > b ) > set_b > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001tf__c,type,
    vimage_b_c: ( b > c ) > set_c > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001tf__d,type,
    vimage_b_d: ( b > d ) > set_d > set_b ).

thf(sy_c_Set_Ovimage_001tf__c_001t__Set__Oset_Itf__b_J,type,
    vimage_c_set_b: ( c > set_b ) > set_set_b > set_c ).

thf(sy_c_Set_Ovimage_001tf__c_001t__Set__Oset_Itf__d_J,type,
    vimage_c_set_d: ( c > set_d ) > set_set_d > set_c ).

thf(sy_c_Set_Ovimage_001tf__c_001tf__b,type,
    vimage_c_b: ( c > b ) > set_b > set_c ).

thf(sy_c_Set_Ovimage_001tf__c_001tf__c,type,
    vimage_c_c: ( c > c ) > set_c > set_c ).

thf(sy_c_Set_Ovimage_001tf__c_001tf__d,type,
    vimage_c_d: ( c > d ) > set_d > set_c ).

thf(sy_c_Set_Ovimage_001tf__d_001t__Set__Oset_Itf__b_J,type,
    vimage_d_set_b: ( d > set_b ) > set_set_b > set_d ).

thf(sy_c_Set_Ovimage_001tf__d_001tf__b,type,
    vimage_d_b: ( d > b ) > set_b > set_d ).

thf(sy_c_Set_Ovimage_001tf__d_001tf__c,type,
    vimage_d_c: ( d > c ) > set_c > set_d ).

thf(sy_c_Set_Ovimage_001tf__d_001tf__d,type,
    vimage_d_d: ( d > d ) > set_d > set_d ).

thf(sy_c_Sigma__Algebra_Oclosed__cdi_001tf__b,type,
    sigma_closed_cdi_b: set_b > set_set_b > $o ).

thf(sy_c_Sigma__Algebra_Oclosed__cdi_001tf__d,type,
    sigma_closed_cdi_d: set_d > set_set_d > $o ).

thf(sy_c_Sigma__Algebra_Oemeasure_001t__Set__Oset_Itf__b_J,type,
    sigma_emeasure_set_b: sigma_measure_set_b > set_set_b > extend8495563244428889912nnreal ).

thf(sy_c_Sigma__Algebra_Oemeasure_001tf__b,type,
    sigma_emeasure_b: sigma_measure_b > set_b > extend8495563244428889912nnreal ).

thf(sy_c_Sigma__Algebra_Oemeasure_001tf__d,type,
    sigma_emeasure_d: sigma_measure_d > set_d > extend8495563244428889912nnreal ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Set__Oset_Itf__b_J_001tf__b,type,
    sigma_3265996224607154480et_b_b: sigma_measure_set_b > sigma_measure_b > set_set_b_b2 ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Set__Oset_Itf__b_J_001tf__d,type,
    sigma_3265996224607154482et_b_d: sigma_measure_set_b > sigma_measure_d > set_set_b_d2 ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Set__Oset_Itf__d_J_001tf__c,type,
    sigma_1994698223397181103et_d_c: sigma_measure_set_d > sigma_measure_c > set_set_d_c ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001t__Set__Oset_Itf__b_J,type,
    sigma_5029598207865514736_set_b: sigma_measure_b > sigma_measure_set_b > set_b_set_b ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__b,type,
    sigma_measurable_b_b: sigma_measure_b > sigma_measure_b > set_b_b ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__c,type,
    sigma_measurable_b_c: sigma_measure_b > sigma_measure_c > set_b_c ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__d,type,
    sigma_measurable_b_d: sigma_measure_b > sigma_measure_d > set_b_d ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001t__Set__Oset_Itf__d_J,type,
    sigma_6374063253281460273_set_d: sigma_measure_c > sigma_measure_set_d > set_c_set_d ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001tf__b,type,
    sigma_measurable_c_b: sigma_measure_c > sigma_measure_b > set_c_b ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001tf__c,type,
    sigma_measurable_c_c: sigma_measure_c > sigma_measure_c > set_c_c ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001tf__d,type,
    sigma_measurable_c_d: sigma_measure_c > sigma_measure_d > set_c_d ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__d_001t__Set__Oset_Itf__b_J,type,
    sigma_7718528281484490606_set_b: sigma_measure_d > sigma_measure_set_b > set_d_set_b ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__d_001tf__b,type,
    sigma_measurable_d_b: sigma_measure_d > sigma_measure_b > set_d_b ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__d_001tf__c,type,
    sigma_measurable_d_c: sigma_measure_d > sigma_measure_c > set_d_c ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__d_001tf__d,type,
    sigma_measurable_d_d: sigma_measure_d > sigma_measure_d > set_d_d ).

thf(sy_c_Sigma__Algebra_Omeasure_001t__Set__Oset_Itf__b_J,type,
    sigma_measure_set_b2: sigma_measure_set_b > set_set_b > real ).

thf(sy_c_Sigma__Algebra_Omeasure_001tf__b,type,
    sigma_measure_b2: sigma_measure_b > set_b > real ).

thf(sy_c_Sigma__Algebra_Omeasure_001tf__d,type,
    sigma_measure_d2: sigma_measure_d > set_d > real ).

thf(sy_c_Sigma__Algebra_Omeasure__of_001t__Set__Oset_Itf__b_J,type,
    sigma_1409593684724707960_set_b: set_set_b > set_set_set_b > ( set_set_b > extend8495563244428889912nnreal ) > sigma_measure_set_b ).

thf(sy_c_Sigma__Algebra_Omeasure__of_001tf__b,type,
    sigma_measure_of_b: set_b > set_set_b > ( set_b > extend8495563244428889912nnreal ) > sigma_measure_b ).

thf(sy_c_Sigma__Algebra_Omeasure__of_001tf__d,type,
    sigma_measure_of_d: set_d > set_set_d > ( set_d > extend8495563244428889912nnreal ) > sigma_measure_d ).

thf(sy_c_Sigma__Algebra_Orestrict__space_001_062_Itf__b_Mtf__b_J,type,
    sigma_547908252458998241ce_b_b: sigma_measure_b_b > set_b_b > sigma_measure_b_b ).

thf(sy_c_Sigma__Algebra_Orestrict__space_001_062_Itf__b_Mtf__d_J,type,
    sigma_547908261065455843ce_b_d: sigma_measure_b_d > set_b_d > sigma_measure_b_d ).

thf(sy_c_Sigma__Algebra_Orestrict__space_001_062_Itf__d_Mtf__b_J,type,
    sigma_4196369125368421983ce_d_b: sigma_measure_d_b > set_d_b > sigma_measure_d_b ).

thf(sy_c_Sigma__Algebra_Orestrict__space_001_062_Itf__d_Mtf__d_J,type,
    sigma_4196369133974879585ce_d_d: sigma_measure_d_d > set_d_d > sigma_measure_d_d ).

thf(sy_c_Sigma__Algebra_Orestrict__space_001t__Set__Oset_Itf__b_J,type,
    sigma_5799942775684512035_set_b: sigma_measure_set_b > set_set_b > sigma_measure_set_b ).

thf(sy_c_Sigma__Algebra_Orestrict__space_001t__Set__Oset_Itf__d_J,type,
    sigma_5799942784290969637_set_d: sigma_measure_set_d > set_set_d > sigma_measure_set_d ).

thf(sy_c_Sigma__Algebra_Orestrict__space_001tf__b,type,
    sigma_8692839461743104067pace_b: sigma_measure_b > set_b > sigma_measure_b ).

thf(sy_c_Sigma__Algebra_Orestrict__space_001tf__c,type,
    sigma_8692839461743104068pace_c: sigma_measure_c > set_c > sigma_measure_c ).

thf(sy_c_Sigma__Algebra_Orestrict__space_001tf__d,type,
    sigma_8692839461743104069pace_d: sigma_measure_d > set_d > sigma_measure_d ).

thf(sy_c_Sigma__Algebra_Osets_001_062_Itf__b_Mtf__b_J,type,
    sigma_sets_b_b: sigma_measure_b_b > set_set_b_b ).

thf(sy_c_Sigma__Algebra_Osets_001_062_Itf__b_Mtf__d_J,type,
    sigma_sets_b_d: sigma_measure_b_d > set_set_b_d ).

thf(sy_c_Sigma__Algebra_Osets_001_062_Itf__d_Mtf__b_J,type,
    sigma_sets_d_b: sigma_measure_d_b > set_set_d_b ).

thf(sy_c_Sigma__Algebra_Osets_001_062_Itf__d_Mtf__d_J,type,
    sigma_sets_d_d: sigma_measure_d_d > set_set_d_d ).

thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_Itf__b_J,type,
    sigma_sets_set_b: sigma_measure_set_b > set_set_set_b ).

thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_Itf__d_J,type,
    sigma_sets_set_d: sigma_measure_set_d > set_set_set_d ).

thf(sy_c_Sigma__Algebra_Osets_001tf__b,type,
    sigma_sets_b: sigma_measure_b > set_set_b ).

thf(sy_c_Sigma__Algebra_Osets_001tf__c,type,
    sigma_sets_c: sigma_measure_c > set_set_c ).

thf(sy_c_Sigma__Algebra_Osets_001tf__d,type,
    sigma_sets_d: sigma_measure_d > set_set_d ).

thf(sy_c_Sigma__Algebra_Osigma__sets_001t__Set__Oset_Itf__b_J,type,
    sigma_2987359972167793591_set_b: set_set_b > set_set_set_b > set_set_set_b ).

thf(sy_c_Sigma__Algebra_Osigma__sets_001tf__b,type,
    sigma_sigma_sets_b: set_b > set_set_b > set_set_b ).

thf(sy_c_Sigma__Algebra_Osigma__sets_001tf__d,type,
    sigma_sigma_sets_d: set_d > set_set_d > set_set_d ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_Itf__b_Mtf__b_J,type,
    sigma_space_b_b: sigma_measure_b_b > set_b_b ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_Itf__b_Mtf__d_J,type,
    sigma_space_b_d: sigma_measure_b_d > set_b_d ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_Itf__d_Mtf__b_J,type,
    sigma_space_d_b: sigma_measure_d_b > set_d_b ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_Itf__d_Mtf__d_J,type,
    sigma_space_d_d: sigma_measure_d_d > set_d_d ).

thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_Itf__b_J,type,
    sigma_space_set_b: sigma_measure_set_b > set_set_b ).

thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_Itf__d_J,type,
    sigma_space_set_d: sigma_measure_set_d > set_set_d ).

thf(sy_c_Sigma__Algebra_Ospace_001tf__b,type,
    sigma_space_b: sigma_measure_b > set_b ).

thf(sy_c_Sigma__Algebra_Ospace_001tf__c,type,
    sigma_space_c: sigma_measure_c > set_c ).

thf(sy_c_Sigma__Algebra_Ospace_001tf__d,type,
    sigma_space_d: sigma_measure_d > set_d ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001_062_Itf__b_Mtf__b_J_001tf__d,type,
    sigma_931312062296759430_b_b_d: set_b_b > ( ( b > b ) > d ) > sigma_measure_d > sigma_measure_b_b ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001_062_Itf__b_Mtf__d_J_001tf__d,type,
    sigma_8883386097941561860_b_d_d: set_b_d > ( ( b > d ) > d ) > sigma_measure_d > sigma_measure_b_d ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001_062_Itf__d_Mtf__b_J_001tf__d,type,
    sigma_2900377255818245000_d_b_d: set_d_b > ( ( d > b ) > d ) > sigma_measure_d > sigma_measure_d_b ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001_062_Itf__d_Mtf__d_J_001tf__d,type,
    sigma_1629079254608271622_d_d_d: set_d_d > ( ( d > d ) > d ) > sigma_measure_d > sigma_measure_d_d ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
    sigma_3488951009927027028_set_b: set_set_b > ( set_b > set_b ) > sigma_measure_set_b > sigma_measure_set_b ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001t__Set__Oset_Itf__b_J_001tf__b,type,
    sigma_6754043411966162036et_b_b: set_set_b > ( set_b > b ) > sigma_measure_b > sigma_measure_set_b ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001t__Set__Oset_Itf__b_J_001tf__d,type,
    sigma_6754043411966162038et_b_d: set_set_b > ( set_b > d ) > sigma_measure_d > sigma_measure_set_b ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001t__Set__Oset_Itf__d_J_001tf__b,type,
    sigma_5482745410756188658et_d_b: set_set_d > ( set_d > b ) > sigma_measure_b > sigma_measure_set_d ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001t__Set__Oset_Itf__d_J_001tf__d,type,
    sigma_5482745410756188660et_d_d: set_set_d > ( set_d > d ) > sigma_measure_d > sigma_measure_set_d ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001tf__b_001tf__b,type,
    sigma_8454314777852479828ra_b_b: set_b > ( b > b ) > sigma_measure_b > sigma_measure_b ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001tf__b_001tf__d,type,
    sigma_8454314777852479830ra_b_d: set_b > ( b > d ) > sigma_measure_d > sigma_measure_b ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001tf__c_001tf__b,type,
    sigma_5666859195879803795ra_c_b: set_c > ( c > b ) > sigma_measure_b > sigma_measure_c ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001tf__c_001tf__d,type,
    sigma_5666859195879803797ra_c_d: set_c > ( c > d ) > sigma_measure_d > sigma_measure_c ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001tf__d_001tf__b,type,
    sigma_2879403613907127762ra_d_b: set_d > ( d > b ) > sigma_measure_b > sigma_measure_d ).

thf(sy_c_Sigma__Algebra_Ovimage__algebra_001tf__d_001tf__d,type,
    sigma_2879403613907127764ra_d_d: set_d > ( d > d ) > sigma_measure_d > sigma_measure_d ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__b_J_Mtf__b_J,type,
    member_set_b_b: ( set_b > b ) > set_set_b_b2 > $o ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__b_J_Mtf__c_J,type,
    member_set_b_c: ( set_b > c ) > set_set_b_c > $o ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__b_J_Mtf__d_J,type,
    member_set_b_d: ( set_b > d ) > set_set_b_d2 > $o ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__d_J_Mt__Set__Oset_Itf__d_J_J,type,
    member_set_d_set_d: ( set_d > set_d ) > set_set_d_set_d > $o ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__d_J_Mtf__c_J,type,
    member_set_d_c: ( set_d > c ) > set_set_d_c > $o ).

thf(sy_c_member_001_062_Itf__b_Mt__Set__Oset_Itf__b_J_J,type,
    member_b_set_b: ( b > set_b ) > set_b_set_b > $o ).

thf(sy_c_member_001_062_Itf__b_Mtf__b_J,type,
    member_b_b: ( b > b ) > set_b_b > $o ).

thf(sy_c_member_001_062_Itf__b_Mtf__c_J,type,
    member_b_c: ( b > c ) > set_b_c > $o ).

thf(sy_c_member_001_062_Itf__b_Mtf__d_J,type,
    member_b_d: ( b > d ) > set_b_d > $o ).

thf(sy_c_member_001_062_Itf__c_Mt__Set__Oset_Itf__b_J_J,type,
    member_c_set_b: ( c > set_b ) > set_c_set_b > $o ).

thf(sy_c_member_001_062_Itf__c_Mt__Set__Oset_Itf__d_J_J,type,
    member_c_set_d: ( c > set_d ) > set_c_set_d > $o ).

thf(sy_c_member_001_062_Itf__c_Mtf__b_J,type,
    member_c_b: ( c > b ) > set_c_b > $o ).

thf(sy_c_member_001_062_Itf__c_Mtf__c_J,type,
    member_c_c: ( c > c ) > set_c_c > $o ).

thf(sy_c_member_001_062_Itf__c_Mtf__d_J,type,
    member_c_d: ( c > d ) > set_c_d > $o ).

thf(sy_c_member_001_062_Itf__d_Mt__Set__Oset_Itf__b_J_J,type,
    member_d_set_b: ( d > set_b ) > set_d_set_b > $o ).

thf(sy_c_member_001_062_Itf__d_Mtf__b_J,type,
    member_d_b: ( d > b ) > set_d_b > $o ).

thf(sy_c_member_001_062_Itf__d_Mtf__c_J,type,
    member_d_c: ( d > c ) > set_d_c > $o ).

thf(sy_c_member_001_062_Itf__d_Mtf__d_J,type,
    member_d_d: ( d > d ) > set_d_d > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
    member_set_b_b2: set_b_b > set_set_b_b > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_Itf__b_Mtf__d_J_J,type,
    member_set_b_d2: set_b_d > set_set_b_d > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_Itf__d_Mtf__b_J_J,type,
    member_set_d_b: set_d_b > set_set_d_b > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_Itf__d_Mtf__d_J_J,type,
    member_set_d_d: set_d_d > set_set_d_d > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    member_set_set_b: set_set_b > set_set_set_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__d_J_J,type,
    member_set_set_d: set_set_d > set_set_set_d > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
    member_set_b: set_b > set_set_b > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__c_J,type,
    member_set_c: set_c > set_set_c > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__d_J,type,
    member_set_d: set_d > set_set_d > $o ).

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_c_member_001tf__c,type,
    member_c: c > set_c > $o ).

thf(sy_c_member_001tf__d,type,
    member_d: d > set_d > $o ).

thf(sy_v_I,type,
    i: set_c ).

thf(sy_v_M_H,type,
    m: c > sigma_measure_d ).

thf(sy_v_N,type,
    n: sigma_measure_b ).

thf(sy_v_X_H,type,
    x: c > b > d ).

thf(sy_v_i____,type,
    i2: c ).

% Relevant facts (1268)
thf(fact_0__092_060open_062i_A_092_060in_062_AI_092_060close_062,axiom,
    member_c @ i2 @ i ).

% \<open>i \<in> I\<close>
thf(fact_1_measurable__sets,axiom,
    ! [F: set_b > d,M: sigma_measure_set_b,A: sigma_measure_d,S: set_d] :
      ( ( member_set_b_d @ F @ ( sigma_3265996224607154482et_b_d @ M @ A ) )
     => ( ( member_set_d @ S @ ( sigma_sets_d @ A ) )
       => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ S ) @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) ) ) ) ).

% measurable_sets
thf(fact_2_measurable__sets,axiom,
    ! [F: set_b > b,M: sigma_measure_set_b,A: sigma_measure_b,S: set_b] :
      ( ( member_set_b_b @ F @ ( sigma_3265996224607154480et_b_b @ M @ A ) )
     => ( ( member_set_b @ S @ ( sigma_sets_b @ A ) )
       => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ S ) @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) ) ) ) ).

% measurable_sets
thf(fact_3_measurable__sets,axiom,
    ! [F: d > d,M: sigma_measure_d,A: sigma_measure_d,S: set_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ A ) )
     => ( ( member_set_d @ S @ ( sigma_sets_d @ A ) )
       => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_d @ F @ S ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) ) ) ).

% measurable_sets
thf(fact_4_measurable__sets,axiom,
    ! [F: d > b,M: sigma_measure_d,A: sigma_measure_b,S: set_b] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ A ) )
     => ( ( member_set_b @ S @ ( sigma_sets_b @ A ) )
       => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_b @ F @ S ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) ) ) ).

% measurable_sets
thf(fact_5_measurable__sets,axiom,
    ! [F: b > b,M: sigma_measure_b,A: sigma_measure_b,S: set_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ A ) )
     => ( ( member_set_b @ S @ ( sigma_sets_b @ A ) )
       => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_b @ F @ S ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_sets
thf(fact_6_measurable__sets,axiom,
    ! [F: b > d,M: sigma_measure_b,A: sigma_measure_d,S: set_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ A ) )
     => ( ( member_set_d @ S @ ( sigma_sets_d @ A ) )
       => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_d @ F @ S ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_sets
thf(fact_7_assms_I2_J,axiom,
    ! [I: c] :
      ( ( member_c @ I @ i )
     => ( member_b_d @ ( x @ I ) @ ( sigma_measurable_b_d @ n @ ( m @ I ) ) ) ) ).

% assms(2)
thf(fact_8_sets_OInt__space__eq1,axiom,
    ! [X: set_set_b,M: sigma_measure_set_b] :
      ( ( member_set_set_b @ X @ ( sigma_sets_set_b @ M ) )
     => ( ( inf_inf_set_set_b @ ( sigma_space_set_b @ M ) @ X )
        = X ) ) ).

% sets.Int_space_eq1
thf(fact_9_sets_OInt__space__eq1,axiom,
    ! [X: set_d,M: sigma_measure_d] :
      ( ( member_set_d @ X @ ( sigma_sets_d @ M ) )
     => ( ( inf_inf_set_d @ ( sigma_space_d @ M ) @ X )
        = X ) ) ).

% sets.Int_space_eq1
thf(fact_10_sets_OInt__space__eq1,axiom,
    ! [X: set_b,M: sigma_measure_b] :
      ( ( member_set_b @ X @ ( sigma_sets_b @ M ) )
     => ( ( inf_inf_set_b @ ( sigma_space_b @ M ) @ X )
        = X ) ) ).

% sets.Int_space_eq1
thf(fact_11_sets_OInt__space__eq2,axiom,
    ! [X: set_set_b,M: sigma_measure_set_b] :
      ( ( member_set_set_b @ X @ ( sigma_sets_set_b @ M ) )
     => ( ( inf_inf_set_set_b @ X @ ( sigma_space_set_b @ M ) )
        = X ) ) ).

% sets.Int_space_eq2
thf(fact_12_sets_OInt__space__eq2,axiom,
    ! [X: set_d,M: sigma_measure_d] :
      ( ( member_set_d @ X @ ( sigma_sets_d @ M ) )
     => ( ( inf_inf_set_d @ X @ ( sigma_space_d @ M ) )
        = X ) ) ).

% sets.Int_space_eq2
thf(fact_13_sets_OInt__space__eq2,axiom,
    ! [X: set_b,M: sigma_measure_b] :
      ( ( member_set_b @ X @ ( sigma_sets_b @ M ) )
     => ( ( inf_inf_set_b @ X @ ( sigma_space_b @ M ) )
        = X ) ) ).

% sets.Int_space_eq2
thf(fact_14_vimage__Int,axiom,
    ! [F: d > b,A: set_b,B: set_b] :
      ( ( vimage_d_b @ F @ ( inf_inf_set_b @ A @ B ) )
      = ( inf_inf_set_d @ ( vimage_d_b @ F @ A ) @ ( vimage_d_b @ F @ B ) ) ) ).

% vimage_Int
thf(fact_15_vimage__Int,axiom,
    ! [F: set_b > b,A: set_b,B: set_b] :
      ( ( vimage_set_b_b @ F @ ( inf_inf_set_b @ A @ B ) )
      = ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ A ) @ ( vimage_set_b_b @ F @ B ) ) ) ).

% vimage_Int
thf(fact_16_vimage__Int,axiom,
    ! [F: d > d,A: set_d,B: set_d] :
      ( ( vimage_d_d @ F @ ( inf_inf_set_d @ A @ B ) )
      = ( inf_inf_set_d @ ( vimage_d_d @ F @ A ) @ ( vimage_d_d @ F @ B ) ) ) ).

% vimage_Int
thf(fact_17_vimage__Int,axiom,
    ! [F: set_b > d,A: set_d,B: set_d] :
      ( ( vimage_set_b_d @ F @ ( inf_inf_set_d @ A @ B ) )
      = ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ A ) @ ( vimage_set_b_d @ F @ B ) ) ) ).

% vimage_Int
thf(fact_18_vimage__Int,axiom,
    ! [F: b > set_b,A: set_set_b,B: set_set_b] :
      ( ( vimage_b_set_b @ F @ ( inf_inf_set_set_b @ A @ B ) )
      = ( inf_inf_set_b @ ( vimage_b_set_b @ F @ A ) @ ( vimage_b_set_b @ F @ B ) ) ) ).

% vimage_Int
thf(fact_19_vimage__Int,axiom,
    ! [F: d > set_b,A: set_set_b,B: set_set_b] :
      ( ( vimage_d_set_b @ F @ ( inf_inf_set_set_b @ A @ B ) )
      = ( inf_inf_set_d @ ( vimage_d_set_b @ F @ A ) @ ( vimage_d_set_b @ F @ B ) ) ) ).

% vimage_Int
thf(fact_20_vimage__Int,axiom,
    ! [F: set_b > set_b,A: set_set_b,B: set_set_b] :
      ( ( vimage_set_b_set_b @ F @ ( inf_inf_set_set_b @ A @ B ) )
      = ( inf_inf_set_set_b @ ( vimage_set_b_set_b @ F @ A ) @ ( vimage_set_b_set_b @ F @ B ) ) ) ).

% vimage_Int
thf(fact_21_vimage__Int,axiom,
    ! [F: b > d,A: set_d,B: set_d] :
      ( ( vimage_b_d @ F @ ( inf_inf_set_d @ A @ B ) )
      = ( inf_inf_set_b @ ( vimage_b_d @ F @ A ) @ ( vimage_b_d @ F @ B ) ) ) ).

% vimage_Int
thf(fact_22_vimage__Int,axiom,
    ! [F: b > b,A: set_b,B: set_b] :
      ( ( vimage_b_b @ F @ ( inf_inf_set_b @ A @ B ) )
      = ( inf_inf_set_b @ ( vimage_b_b @ F @ A ) @ ( vimage_b_b @ F @ B ) ) ) ).

% vimage_Int
thf(fact_23_sets_Otop,axiom,
    ! [M: sigma_measure_d] : ( member_set_d @ ( sigma_space_d @ M ) @ ( sigma_sets_d @ M ) ) ).

% sets.top
thf(fact_24_sets_Otop,axiom,
    ! [M: sigma_measure_b] : ( member_set_b @ ( sigma_space_b @ M ) @ ( sigma_sets_b @ M ) ) ).

% sets.top
thf(fact_25_sets_OInt,axiom,
    ! [A2: set_set_b,M: sigma_measure_set_b,B2: set_set_b] :
      ( ( member_set_set_b @ A2 @ ( sigma_sets_set_b @ M ) )
     => ( ( member_set_set_b @ B2 @ ( sigma_sets_set_b @ M ) )
       => ( member_set_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ ( sigma_sets_set_b @ M ) ) ) ) ).

% sets.Int
thf(fact_26_sets_OInt,axiom,
    ! [A2: set_d,M: sigma_measure_d,B2: set_d] :
      ( ( member_set_d @ A2 @ ( sigma_sets_d @ M ) )
     => ( ( member_set_d @ B2 @ ( sigma_sets_d @ M ) )
       => ( member_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ ( sigma_sets_d @ M ) ) ) ) ).

% sets.Int
thf(fact_27_sets_OInt,axiom,
    ! [A2: set_b,M: sigma_measure_b,B2: set_b] :
      ( ( member_set_b @ A2 @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b @ B2 @ ( sigma_sets_b @ M ) )
       => ( member_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( sigma_sets_b @ M ) ) ) ) ).

% sets.Int
thf(fact_28_Int__subset__iff,axiom,
    ! [C: set_d,A: set_d,B: set_d] :
      ( ( ord_less_eq_set_d @ C @ ( inf_inf_set_d @ A @ B ) )
      = ( ( ord_less_eq_set_d @ C @ A )
        & ( ord_less_eq_set_d @ C @ B ) ) ) ).

% Int_subset_iff
thf(fact_29_Int__subset__iff,axiom,
    ! [C: set_b,A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ C @ ( inf_inf_set_b @ A @ B ) )
      = ( ( ord_less_eq_set_b @ C @ A )
        & ( ord_less_eq_set_b @ C @ B ) ) ) ).

% Int_subset_iff
thf(fact_30_Int__subset__iff,axiom,
    ! [C: set_set_b,A: set_set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) )
      = ( ( ord_le3795704787696855135_set_b @ C @ A )
        & ( ord_le3795704787696855135_set_b @ C @ B ) ) ) ).

% Int_subset_iff
thf(fact_31_le__inf__iff,axiom,
    ! [X: set_d,Y: set_d,Z: set_d] :
      ( ( ord_less_eq_set_d @ X @ ( inf_inf_set_d @ Y @ Z ) )
      = ( ( ord_less_eq_set_d @ X @ Y )
        & ( ord_less_eq_set_d @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_32_le__inf__iff,axiom,
    ! [X: set_b,Y: set_b,Z: set_b] :
      ( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
      = ( ( ord_less_eq_set_b @ X @ Y )
        & ( ord_less_eq_set_b @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_33_le__inf__iff,axiom,
    ! [X: set_set_b,Y: set_set_b,Z: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ X @ ( inf_inf_set_set_b @ Y @ Z ) )
      = ( ( ord_le3795704787696855135_set_b @ X @ Y )
        & ( ord_le3795704787696855135_set_b @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_34_inf_Obounded__iff,axiom,
    ! [A2: set_d,B2: set_d,C2: set_d] :
      ( ( ord_less_eq_set_d @ A2 @ ( inf_inf_set_d @ B2 @ C2 ) )
      = ( ( ord_less_eq_set_d @ A2 @ B2 )
        & ( ord_less_eq_set_d @ A2 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_35_inf_Obounded__iff,axiom,
    ! [A2: set_set_b,B2: set_set_b,C2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ ( inf_inf_set_set_b @ B2 @ C2 ) )
      = ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
        & ( ord_le3795704787696855135_set_b @ A2 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_36_inf_Obounded__iff,axiom,
    ! [A2: set_b,B2: set_b,C2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C2 ) )
      = ( ( ord_less_eq_set_b @ A2 @ B2 )
        & ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_37_sets__le__imp__space__le,axiom,
    ! [A: sigma_measure_d,B: sigma_measure_d] :
      ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ A ) @ ( sigma_sets_d @ B ) )
     => ( ord_less_eq_set_d @ ( sigma_space_d @ A ) @ ( sigma_space_d @ B ) ) ) ).

% sets_le_imp_space_le
thf(fact_38_sets__le__imp__space__le,axiom,
    ! [A: sigma_measure_set_b,B: sigma_measure_set_b] :
      ( ( ord_le3201067847557142847_set_b @ ( sigma_sets_set_b @ A ) @ ( sigma_sets_set_b @ B ) )
     => ( ord_le3795704787696855135_set_b @ ( sigma_space_set_b @ A ) @ ( sigma_space_set_b @ B ) ) ) ).

% sets_le_imp_space_le
thf(fact_39_sets__le__imp__space__le,axiom,
    ! [A: sigma_measure_b,B: sigma_measure_b] :
      ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ A ) @ ( sigma_sets_b @ B ) )
     => ( ord_less_eq_set_b @ ( sigma_space_b @ A ) @ ( sigma_space_b @ B ) ) ) ).

% sets_le_imp_space_le
thf(fact_40_vimage__ident,axiom,
    ! [Y2: set_b] :
      ( ( vimage_b_b
        @ ^ [X2: b] : X2
        @ Y2 )
      = Y2 ) ).

% vimage_ident
thf(fact_41_vimage__ident,axiom,
    ! [Y2: set_d] :
      ( ( vimage_d_d
        @ ^ [X2: d] : X2
        @ Y2 )
      = Y2 ) ).

% vimage_ident
thf(fact_42_vimage__Collect__eq,axiom,
    ! [F: b > d,P: d > $o] :
      ( ( vimage_b_d @ F @ ( collect_d @ P ) )
      = ( collect_b
        @ ^ [Y3: b] : ( P @ ( F @ Y3 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_43_vimage__Collect__eq,axiom,
    ! [F: b > b,P: b > $o] :
      ( ( vimage_b_b @ F @ ( collect_b @ P ) )
      = ( collect_b
        @ ^ [Y3: b] : ( P @ ( F @ Y3 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_44_vimage__Collect__eq,axiom,
    ! [F: d > b,P: b > $o] :
      ( ( vimage_d_b @ F @ ( collect_b @ P ) )
      = ( collect_d
        @ ^ [Y3: d] : ( P @ ( F @ Y3 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_45_vimage__Collect__eq,axiom,
    ! [F: d > d,P: d > $o] :
      ( ( vimage_d_d @ F @ ( collect_d @ P ) )
      = ( collect_d
        @ ^ [Y3: d] : ( P @ ( F @ Y3 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_46_vimage__Collect__eq,axiom,
    ! [F: set_b > set_b,P: set_b > $o] :
      ( ( vimage_set_b_set_b @ F @ ( collect_set_b @ P ) )
      = ( collect_set_b
        @ ^ [Y3: set_b] : ( P @ ( F @ Y3 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_47_subset__antisym,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ( ord_le3795704787696855135_set_b @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_48_subset__antisym,axiom,
    ! [A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( ord_less_eq_set_b @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_49_subsetI,axiom,
    ! [A: set_set_d,B: set_set_d] :
      ( ! [X3: set_d] :
          ( ( member_set_d @ X3 @ A )
         => ( member_set_d @ X3 @ B ) )
     => ( ord_le3937772867789547105_set_d @ A @ B ) ) ).

% subsetI
thf(fact_50_subsetI,axiom,
    ! [A: set_c,B: set_c] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_c @ X3 @ B ) )
     => ( ord_less_eq_set_c @ A @ B ) ) ).

% subsetI
thf(fact_51_subsetI,axiom,
    ! [A: set_b_d,B: set_b_d] :
      ( ! [X3: b > d] :
          ( ( member_b_d @ X3 @ A )
         => ( member_b_d @ X3 @ B ) )
     => ( ord_less_eq_set_b_d @ A @ B ) ) ).

% subsetI
thf(fact_52_subsetI,axiom,
    ! [A: set_b_b,B: set_b_b] :
      ( ! [X3: b > b] :
          ( ( member_b_b @ X3 @ A )
         => ( member_b_b @ X3 @ B ) )
     => ( ord_less_eq_set_b_b @ A @ B ) ) ).

% subsetI
thf(fact_53_subsetI,axiom,
    ! [A: set_d_b,B: set_d_b] :
      ( ! [X3: d > b] :
          ( ( member_d_b @ X3 @ A )
         => ( member_d_b @ X3 @ B ) )
     => ( ord_less_eq_set_d_b @ A @ B ) ) ).

% subsetI
thf(fact_54_subsetI,axiom,
    ! [A: set_d_d,B: set_d_d] :
      ( ! [X3: d > d] :
          ( ( member_d_d @ X3 @ A )
         => ( member_d_d @ X3 @ B ) )
     => ( ord_less_eq_set_d_d @ A @ B ) ) ).

% subsetI
thf(fact_55_subsetI,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ! [X3: set_b] :
          ( ( member_set_b @ X3 @ A )
         => ( member_set_b @ X3 @ B ) )
     => ( ord_le3795704787696855135_set_b @ A @ B ) ) ).

% subsetI
thf(fact_56_subsetI,axiom,
    ! [A: set_b,B: set_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_b @ X3 @ B ) )
     => ( ord_less_eq_set_b @ A @ B ) ) ).

% subsetI
thf(fact_57_inf__right__idem,axiom,
    ! [X: set_b,Y: set_b] :
      ( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y )
      = ( inf_inf_set_b @ X @ Y ) ) ).

% inf_right_idem
thf(fact_58_inf__right__idem,axiom,
    ! [X: set_d,Y: set_d] :
      ( ( inf_inf_set_d @ ( inf_inf_set_d @ X @ Y ) @ Y )
      = ( inf_inf_set_d @ X @ Y ) ) ).

% inf_right_idem
thf(fact_59_inf__right__idem,axiom,
    ! [X: set_set_b,Y: set_set_b] :
      ( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ X @ Y ) @ Y )
      = ( inf_inf_set_set_b @ X @ Y ) ) ).

% inf_right_idem
thf(fact_60_inf_Oright__idem,axiom,
    ! [A2: set_b,B2: set_b] :
      ( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 )
      = ( inf_inf_set_b @ A2 @ B2 ) ) ).

% inf.right_idem
thf(fact_61_inf_Oright__idem,axiom,
    ! [A2: set_d,B2: set_d] :
      ( ( inf_inf_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ B2 )
      = ( inf_inf_set_d @ A2 @ B2 ) ) ).

% inf.right_idem
thf(fact_62_inf_Oright__idem,axiom,
    ! [A2: set_set_b,B2: set_set_b] :
      ( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ B2 )
      = ( inf_inf_set_set_b @ A2 @ B2 ) ) ).

% inf.right_idem
thf(fact_63_inf__left__idem,axiom,
    ! [X: set_b,Y: set_b] :
      ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
      = ( inf_inf_set_b @ X @ Y ) ) ).

% inf_left_idem
thf(fact_64_inf__left__idem,axiom,
    ! [X: set_d,Y: set_d] :
      ( ( inf_inf_set_d @ X @ ( inf_inf_set_d @ X @ Y ) )
      = ( inf_inf_set_d @ X @ Y ) ) ).

% inf_left_idem
thf(fact_65_inf__left__idem,axiom,
    ! [X: set_set_b,Y: set_set_b] :
      ( ( inf_inf_set_set_b @ X @ ( inf_inf_set_set_b @ X @ Y ) )
      = ( inf_inf_set_set_b @ X @ Y ) ) ).

% inf_left_idem
thf(fact_66_inf_Oleft__idem,axiom,
    ! [A2: set_b,B2: set_b] :
      ( ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ A2 @ B2 ) )
      = ( inf_inf_set_b @ A2 @ B2 ) ) ).

% inf.left_idem
thf(fact_67_inf_Oleft__idem,axiom,
    ! [A2: set_d,B2: set_d] :
      ( ( inf_inf_set_d @ A2 @ ( inf_inf_set_d @ A2 @ B2 ) )
      = ( inf_inf_set_d @ A2 @ B2 ) ) ).

% inf.left_idem
thf(fact_68_inf_Oleft__idem,axiom,
    ! [A2: set_set_b,B2: set_set_b] :
      ( ( inf_inf_set_set_b @ A2 @ ( inf_inf_set_set_b @ A2 @ B2 ) )
      = ( inf_inf_set_set_b @ A2 @ B2 ) ) ).

% inf.left_idem
thf(fact_69_inf__idem,axiom,
    ! [X: set_b] :
      ( ( inf_inf_set_b @ X @ X )
      = X ) ).

% inf_idem
thf(fact_70_inf__idem,axiom,
    ! [X: set_d] :
      ( ( inf_inf_set_d @ X @ X )
      = X ) ).

% inf_idem
thf(fact_71_inf__idem,axiom,
    ! [X: set_set_b] :
      ( ( inf_inf_set_set_b @ X @ X )
      = X ) ).

% inf_idem
thf(fact_72_inf_Oidem,axiom,
    ! [A2: set_b] :
      ( ( inf_inf_set_b @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_73_inf_Oidem,axiom,
    ! [A2: set_d] :
      ( ( inf_inf_set_d @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_74_inf_Oidem,axiom,
    ! [A2: set_set_b] :
      ( ( inf_inf_set_set_b @ A2 @ A2 )
      = A2 ) ).

% inf.idem
thf(fact_75_Int__iff,axiom,
    ! [C2: set_d,A: set_set_d,B: set_set_d] :
      ( ( member_set_d @ C2 @ ( inf_inf_set_set_d @ A @ B ) )
      = ( ( member_set_d @ C2 @ A )
        & ( member_set_d @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_76_Int__iff,axiom,
    ! [C2: c,A: set_c,B: set_c] :
      ( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B ) )
      = ( ( member_c @ C2 @ A )
        & ( member_c @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_77_Int__iff,axiom,
    ! [C2: b > d,A: set_b_d,B: set_b_d] :
      ( ( member_b_d @ C2 @ ( inf_inf_set_b_d @ A @ B ) )
      = ( ( member_b_d @ C2 @ A )
        & ( member_b_d @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_78_Int__iff,axiom,
    ! [C2: b > b,A: set_b_b,B: set_b_b] :
      ( ( member_b_b @ C2 @ ( inf_inf_set_b_b @ A @ B ) )
      = ( ( member_b_b @ C2 @ A )
        & ( member_b_b @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_79_Int__iff,axiom,
    ! [C2: d > b,A: set_d_b,B: set_d_b] :
      ( ( member_d_b @ C2 @ ( inf_inf_set_d_b @ A @ B ) )
      = ( ( member_d_b @ C2 @ A )
        & ( member_d_b @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_80_Int__iff,axiom,
    ! [C2: d > d,A: set_d_d,B: set_d_d] :
      ( ( member_d_d @ C2 @ ( inf_inf_set_d_d @ A @ B ) )
      = ( ( member_d_d @ C2 @ A )
        & ( member_d_d @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_81_Int__iff,axiom,
    ! [C2: b,A: set_b,B: set_b] :
      ( ( member_b @ C2 @ ( inf_inf_set_b @ A @ B ) )
      = ( ( member_b @ C2 @ A )
        & ( member_b @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_82_Int__iff,axiom,
    ! [C2: d,A: set_d,B: set_d] :
      ( ( member_d @ C2 @ ( inf_inf_set_d @ A @ B ) )
      = ( ( member_d @ C2 @ A )
        & ( member_d @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_83_Int__iff,axiom,
    ! [C2: set_b,A: set_set_b,B: set_set_b] :
      ( ( member_set_b @ C2 @ ( inf_inf_set_set_b @ A @ B ) )
      = ( ( member_set_b @ C2 @ A )
        & ( member_set_b @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_84_IntI,axiom,
    ! [C2: set_d,A: set_set_d,B: set_set_d] :
      ( ( member_set_d @ C2 @ A )
     => ( ( member_set_d @ C2 @ B )
       => ( member_set_d @ C2 @ ( inf_inf_set_set_d @ A @ B ) ) ) ) ).

% IntI
thf(fact_85_IntI,axiom,
    ! [C2: c,A: set_c,B: set_c] :
      ( ( member_c @ C2 @ A )
     => ( ( member_c @ C2 @ B )
       => ( member_c @ C2 @ ( inf_inf_set_c @ A @ B ) ) ) ) ).

% IntI
thf(fact_86_IntI,axiom,
    ! [C2: b > d,A: set_b_d,B: set_b_d] :
      ( ( member_b_d @ C2 @ A )
     => ( ( member_b_d @ C2 @ B )
       => ( member_b_d @ C2 @ ( inf_inf_set_b_d @ A @ B ) ) ) ) ).

% IntI
thf(fact_87_IntI,axiom,
    ! [C2: b > b,A: set_b_b,B: set_b_b] :
      ( ( member_b_b @ C2 @ A )
     => ( ( member_b_b @ C2 @ B )
       => ( member_b_b @ C2 @ ( inf_inf_set_b_b @ A @ B ) ) ) ) ).

% IntI
thf(fact_88_IntI,axiom,
    ! [C2: d > b,A: set_d_b,B: set_d_b] :
      ( ( member_d_b @ C2 @ A )
     => ( ( member_d_b @ C2 @ B )
       => ( member_d_b @ C2 @ ( inf_inf_set_d_b @ A @ B ) ) ) ) ).

% IntI
thf(fact_89_IntI,axiom,
    ! [C2: d > d,A: set_d_d,B: set_d_d] :
      ( ( member_d_d @ C2 @ A )
     => ( ( member_d_d @ C2 @ B )
       => ( member_d_d @ C2 @ ( inf_inf_set_d_d @ A @ B ) ) ) ) ).

% IntI
thf(fact_90_IntI,axiom,
    ! [C2: b,A: set_b,B: set_b] :
      ( ( member_b @ C2 @ A )
     => ( ( member_b @ C2 @ B )
       => ( member_b @ C2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ).

% IntI
thf(fact_91_IntI,axiom,
    ! [C2: d,A: set_d,B: set_d] :
      ( ( member_d @ C2 @ A )
     => ( ( member_d @ C2 @ B )
       => ( member_d @ C2 @ ( inf_inf_set_d @ A @ B ) ) ) ) ).

% IntI
thf(fact_92_IntI,axiom,
    ! [C2: set_b,A: set_set_b,B: set_set_b] :
      ( ( member_set_b @ C2 @ A )
     => ( ( member_set_b @ C2 @ B )
       => ( member_set_b @ C2 @ ( inf_inf_set_set_b @ A @ B ) ) ) ) ).

% IntI
thf(fact_93_vimage__eq,axiom,
    ! [A2: c,F: c > c,B: set_c] :
      ( ( member_c @ A2 @ ( vimage_c_c @ F @ B ) )
      = ( member_c @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_94_vimage__eq,axiom,
    ! [A2: b,F: b > d,B: set_d] :
      ( ( member_b @ A2 @ ( vimage_b_d @ F @ B ) )
      = ( member_d @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_95_vimage__eq,axiom,
    ! [A2: b,F: b > b,B: set_b] :
      ( ( member_b @ A2 @ ( vimage_b_b @ F @ B ) )
      = ( member_b @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_96_vimage__eq,axiom,
    ! [A2: d,F: d > b,B: set_b] :
      ( ( member_d @ A2 @ ( vimage_d_b @ F @ B ) )
      = ( member_b @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_97_vimage__eq,axiom,
    ! [A2: d,F: d > d,B: set_d] :
      ( ( member_d @ A2 @ ( vimage_d_d @ F @ B ) )
      = ( member_d @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_98_vimage__eq,axiom,
    ! [A2: set_d,F: set_d > c,B: set_c] :
      ( ( member_set_d @ A2 @ ( vimage_set_d_c @ F @ B ) )
      = ( member_c @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_99_vimage__eq,axiom,
    ! [A2: c,F: c > set_d,B: set_set_d] :
      ( ( member_c @ A2 @ ( vimage_c_set_d @ F @ B ) )
      = ( member_set_d @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_100_vimage__eq,axiom,
    ! [A2: c,F: c > set_b,B: set_set_b] :
      ( ( member_c @ A2 @ ( vimage_c_set_b @ F @ B ) )
      = ( member_set_b @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_101_vimage__eq,axiom,
    ! [A2: set_b,F: set_b > c,B: set_c] :
      ( ( member_set_b @ A2 @ ( vimage_set_b_c @ F @ B ) )
      = ( member_c @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_102_vimage__eq,axiom,
    ! [A2: set_d,F: set_d > set_d,B: set_set_d] :
      ( ( member_set_d @ A2 @ ( vimage_set_d_set_d @ F @ B ) )
      = ( member_set_d @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_103_vimageI,axiom,
    ! [F: c > c,A2: c,B2: c,B: set_c] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_c @ B2 @ B )
       => ( member_c @ A2 @ ( vimage_c_c @ F @ B ) ) ) ) ).

% vimageI
thf(fact_104_vimageI,axiom,
    ! [F: b > d,A2: b,B2: d,B: set_d] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_d @ B2 @ B )
       => ( member_b @ A2 @ ( vimage_b_d @ F @ B ) ) ) ) ).

% vimageI
thf(fact_105_vimageI,axiom,
    ! [F: b > b,A2: b,B2: b,B: set_b] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_b @ B2 @ B )
       => ( member_b @ A2 @ ( vimage_b_b @ F @ B ) ) ) ) ).

% vimageI
thf(fact_106_vimageI,axiom,
    ! [F: d > b,A2: d,B2: b,B: set_b] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_b @ B2 @ B )
       => ( member_d @ A2 @ ( vimage_d_b @ F @ B ) ) ) ) ).

% vimageI
thf(fact_107_vimageI,axiom,
    ! [F: d > d,A2: d,B2: d,B: set_d] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_d @ B2 @ B )
       => ( member_d @ A2 @ ( vimage_d_d @ F @ B ) ) ) ) ).

% vimageI
thf(fact_108_vimageI,axiom,
    ! [F: c > set_d,A2: c,B2: set_d,B: set_set_d] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_set_d @ B2 @ B )
       => ( member_c @ A2 @ ( vimage_c_set_d @ F @ B ) ) ) ) ).

% vimageI
thf(fact_109_vimageI,axiom,
    ! [F: set_d > c,A2: set_d,B2: c,B: set_c] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_c @ B2 @ B )
       => ( member_set_d @ A2 @ ( vimage_set_d_c @ F @ B ) ) ) ) ).

% vimageI
thf(fact_110_vimageI,axiom,
    ! [F: set_b > c,A2: set_b,B2: c,B: set_c] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_c @ B2 @ B )
       => ( member_set_b @ A2 @ ( vimage_set_b_c @ F @ B ) ) ) ) ).

% vimageI
thf(fact_111_vimageI,axiom,
    ! [F: c > set_b,A2: c,B2: set_b,B: set_set_b] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_set_b @ B2 @ B )
       => ( member_c @ A2 @ ( vimage_c_set_b @ F @ B ) ) ) ) ).

% vimageI
thf(fact_112_vimageI,axiom,
    ! [F: set_d > set_d,A2: set_d,B2: set_d,B: set_set_d] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_set_d @ B2 @ B )
       => ( member_set_d @ A2 @ ( vimage_set_d_set_d @ F @ B ) ) ) ) ).

% vimageI
thf(fact_113_measurable__compose__rev,axiom,
    ! [F: b > d,L: sigma_measure_b,N: sigma_measure_d,G: b > b,M: sigma_measure_b] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ L @ N ) )
     => ( ( member_b_b @ G @ ( sigma_measurable_b_b @ M @ L ) )
       => ( member_b_d
          @ ^ [X2: b] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_b_d @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_114_measurable__compose__rev,axiom,
    ! [F: b > d,L: sigma_measure_b,N: sigma_measure_d,G: d > b,M: sigma_measure_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ L @ N ) )
     => ( ( member_d_b @ G @ ( sigma_measurable_d_b @ M @ L ) )
       => ( member_d_d
          @ ^ [X2: d] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_d_d @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_115_measurable__compose__rev,axiom,
    ! [F: b > b,L: sigma_measure_b,N: sigma_measure_b,G: b > b,M: sigma_measure_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ L @ N ) )
     => ( ( member_b_b @ G @ ( sigma_measurable_b_b @ M @ L ) )
       => ( member_b_b
          @ ^ [X2: b] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_b_b @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_116_measurable__compose__rev,axiom,
    ! [F: b > b,L: sigma_measure_b,N: sigma_measure_b,G: d > b,M: sigma_measure_d] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ L @ N ) )
     => ( ( member_d_b @ G @ ( sigma_measurable_d_b @ M @ L ) )
       => ( member_d_b
          @ ^ [X2: d] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_d_b @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_117_measurable__compose__rev,axiom,
    ! [F: d > b,L: sigma_measure_d,N: sigma_measure_b,G: b > d,M: sigma_measure_b] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ L @ N ) )
     => ( ( member_b_d @ G @ ( sigma_measurable_b_d @ M @ L ) )
       => ( member_b_b
          @ ^ [X2: b] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_b_b @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_118_measurable__compose__rev,axiom,
    ! [F: d > b,L: sigma_measure_d,N: sigma_measure_b,G: d > d,M: sigma_measure_d] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ L @ N ) )
     => ( ( member_d_d @ G @ ( sigma_measurable_d_d @ M @ L ) )
       => ( member_d_b
          @ ^ [X2: d] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_d_b @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_119_measurable__compose__rev,axiom,
    ! [F: d > d,L: sigma_measure_d,N: sigma_measure_d,G: b > d,M: sigma_measure_b] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ L @ N ) )
     => ( ( member_b_d @ G @ ( sigma_measurable_b_d @ M @ L ) )
       => ( member_b_d
          @ ^ [X2: b] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_b_d @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_120_measurable__compose__rev,axiom,
    ! [F: d > d,L: sigma_measure_d,N: sigma_measure_d,G: d > d,M: sigma_measure_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ L @ N ) )
     => ( ( member_d_d @ G @ ( sigma_measurable_d_d @ M @ L ) )
       => ( member_d_d
          @ ^ [X2: d] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_d_d @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_121_measurable__compose,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d,G: d > b,L: sigma_measure_b] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
     => ( ( member_d_b @ G @ ( sigma_measurable_d_b @ N @ L ) )
       => ( member_b_b
          @ ^ [X2: b] : ( G @ ( F @ X2 ) )
          @ ( sigma_measurable_b_b @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_122_measurable__compose,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d,G: d > d,L: sigma_measure_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
     => ( ( member_d_d @ G @ ( sigma_measurable_d_d @ N @ L ) )
       => ( member_b_d
          @ ^ [X2: b] : ( G @ ( F @ X2 ) )
          @ ( sigma_measurable_b_d @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_123_measurable__compose,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,G: b > d,L: sigma_measure_d] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( ( member_b_d @ G @ ( sigma_measurable_b_d @ N @ L ) )
       => ( member_b_d
          @ ^ [X2: b] : ( G @ ( F @ X2 ) )
          @ ( sigma_measurable_b_d @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_124_measurable__compose,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,G: b > b,L: sigma_measure_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( ( member_b_b @ G @ ( sigma_measurable_b_b @ N @ L ) )
       => ( member_b_b
          @ ^ [X2: b] : ( G @ ( F @ X2 ) )
          @ ( sigma_measurable_b_b @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_125_measurable__compose,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b,G: b > d,L: sigma_measure_d] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
     => ( ( member_b_d @ G @ ( sigma_measurable_b_d @ N @ L ) )
       => ( member_d_d
          @ ^ [X2: d] : ( G @ ( F @ X2 ) )
          @ ( sigma_measurable_d_d @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_126_measurable__compose,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b,G: b > b,L: sigma_measure_b] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
     => ( ( member_b_b @ G @ ( sigma_measurable_b_b @ N @ L ) )
       => ( member_d_b
          @ ^ [X2: d] : ( G @ ( F @ X2 ) )
          @ ( sigma_measurable_d_b @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_127_measurable__compose,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d,G: d > b,L: sigma_measure_b] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
     => ( ( member_d_b @ G @ ( sigma_measurable_d_b @ N @ L ) )
       => ( member_d_b
          @ ^ [X2: d] : ( G @ ( F @ X2 ) )
          @ ( sigma_measurable_d_b @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_128_measurable__compose,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d,G: d > d,L: sigma_measure_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
     => ( ( member_d_d @ G @ ( sigma_measurable_d_d @ N @ L ) )
       => ( member_d_d
          @ ^ [X2: d] : ( G @ ( F @ X2 ) )
          @ ( sigma_measurable_d_d @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_129_measurable__id,axiom,
    ! [M: sigma_measure_b] :
      ( member_b_b
      @ ^ [X2: b] : X2
      @ ( sigma_measurable_b_b @ M @ M ) ) ).

% measurable_id
thf(fact_130_measurable__id,axiom,
    ! [M: sigma_measure_d] :
      ( member_d_d
      @ ^ [X2: d] : X2
      @ ( sigma_measurable_d_d @ M @ M ) ) ).

% measurable_id
thf(fact_131_inf__set__def,axiom,
    ( inf_inf_set_set_d
    = ( ^ [A3: set_set_d,B3: set_set_d] :
          ( collect_set_d
          @ ( inf_inf_set_d_o
            @ ^ [X2: set_d] : ( member_set_d @ X2 @ A3 )
            @ ^ [X2: set_d] : ( member_set_d @ X2 @ B3 ) ) ) ) ) ).

% inf_set_def
thf(fact_132_inf__set__def,axiom,
    ( inf_inf_set_c
    = ( ^ [A3: set_c,B3: set_c] :
          ( collect_c
          @ ( inf_inf_c_o
            @ ^ [X2: c] : ( member_c @ X2 @ A3 )
            @ ^ [X2: c] : ( member_c @ X2 @ B3 ) ) ) ) ) ).

% inf_set_def
thf(fact_133_inf__set__def,axiom,
    ( inf_inf_set_b_d
    = ( ^ [A3: set_b_d,B3: set_b_d] :
          ( collect_b_d
          @ ( inf_inf_b_d_o
            @ ^ [X2: b > d] : ( member_b_d @ X2 @ A3 )
            @ ^ [X2: b > d] : ( member_b_d @ X2 @ B3 ) ) ) ) ) ).

% inf_set_def
thf(fact_134_inf__set__def,axiom,
    ( inf_inf_set_b_b
    = ( ^ [A3: set_b_b,B3: set_b_b] :
          ( collect_b_b
          @ ( inf_inf_b_b_o
            @ ^ [X2: b > b] : ( member_b_b @ X2 @ A3 )
            @ ^ [X2: b > b] : ( member_b_b @ X2 @ B3 ) ) ) ) ) ).

% inf_set_def
thf(fact_135_inf__set__def,axiom,
    ( inf_inf_set_d_b
    = ( ^ [A3: set_d_b,B3: set_d_b] :
          ( collect_d_b
          @ ( inf_inf_d_b_o
            @ ^ [X2: d > b] : ( member_d_b @ X2 @ A3 )
            @ ^ [X2: d > b] : ( member_d_b @ X2 @ B3 ) ) ) ) ) ).

% inf_set_def
thf(fact_136_inf__set__def,axiom,
    ( inf_inf_set_d_d
    = ( ^ [A3: set_d_d,B3: set_d_d] :
          ( collect_d_d
          @ ( inf_inf_d_d_o
            @ ^ [X2: d > d] : ( member_d_d @ X2 @ A3 )
            @ ^ [X2: d > d] : ( member_d_d @ X2 @ B3 ) ) ) ) ) ).

% inf_set_def
thf(fact_137_inf__set__def,axiom,
    ( inf_inf_set_b
    = ( ^ [A3: set_b,B3: set_b] :
          ( collect_b
          @ ( inf_inf_b_o
            @ ^ [X2: b] : ( member_b @ X2 @ A3 )
            @ ^ [X2: b] : ( member_b @ X2 @ B3 ) ) ) ) ) ).

% inf_set_def
thf(fact_138_inf__set__def,axiom,
    ( inf_inf_set_d
    = ( ^ [A3: set_d,B3: set_d] :
          ( collect_d
          @ ( inf_inf_d_o
            @ ^ [X2: d] : ( member_d @ X2 @ A3 )
            @ ^ [X2: d] : ( member_d @ X2 @ B3 ) ) ) ) ) ).

% inf_set_def
thf(fact_139_inf__set__def,axiom,
    ( inf_inf_set_set_b
    = ( ^ [A3: set_set_b,B3: set_set_b] :
          ( collect_set_b
          @ ( inf_inf_set_b_o
            @ ^ [X2: set_b] : ( member_set_b @ X2 @ A3 )
            @ ^ [X2: set_b] : ( member_set_b @ X2 @ B3 ) ) ) ) ) ).

% inf_set_def
thf(fact_140_less__eq__set__def,axiom,
    ( ord_le3937772867789547105_set_d
    = ( ^ [A3: set_set_d,B3: set_set_d] :
          ( ord_less_eq_set_d_o
          @ ^ [X2: set_d] : ( member_set_d @ X2 @ A3 )
          @ ^ [X2: set_d] : ( member_set_d @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_141_less__eq__set__def,axiom,
    ( ord_less_eq_set_c
    = ( ^ [A3: set_c,B3: set_c] :
          ( ord_less_eq_c_o
          @ ^ [X2: c] : ( member_c @ X2 @ A3 )
          @ ^ [X2: c] : ( member_c @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_142_less__eq__set__def,axiom,
    ( ord_less_eq_set_b_d
    = ( ^ [A3: set_b_d,B3: set_b_d] :
          ( ord_less_eq_b_d_o
          @ ^ [X2: b > d] : ( member_b_d @ X2 @ A3 )
          @ ^ [X2: b > d] : ( member_b_d @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_143_less__eq__set__def,axiom,
    ( ord_less_eq_set_b_b
    = ( ^ [A3: set_b_b,B3: set_b_b] :
          ( ord_less_eq_b_b_o
          @ ^ [X2: b > b] : ( member_b_b @ X2 @ A3 )
          @ ^ [X2: b > b] : ( member_b_b @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_144_less__eq__set__def,axiom,
    ( ord_less_eq_set_d_b
    = ( ^ [A3: set_d_b,B3: set_d_b] :
          ( ord_less_eq_d_b_o
          @ ^ [X2: d > b] : ( member_d_b @ X2 @ A3 )
          @ ^ [X2: d > b] : ( member_d_b @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_145_less__eq__set__def,axiom,
    ( ord_less_eq_set_d_d
    = ( ^ [A3: set_d_d,B3: set_d_d] :
          ( ord_less_eq_d_d_o
          @ ^ [X2: d > d] : ( member_d_d @ X2 @ A3 )
          @ ^ [X2: d > d] : ( member_d_d @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_146_less__eq__set__def,axiom,
    ( ord_le3795704787696855135_set_b
    = ( ^ [A3: set_set_b,B3: set_set_b] :
          ( ord_less_eq_set_b_o
          @ ^ [X2: set_b] : ( member_set_b @ X2 @ A3 )
          @ ^ [X2: set_b] : ( member_set_b @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_147_less__eq__set__def,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A3: set_b,B3: set_b] :
          ( ord_less_eq_b_o
          @ ^ [X2: b] : ( member_b @ X2 @ A3 )
          @ ^ [X2: b] : ( member_b @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_148_measurable__cong__sets,axiom,
    ! [M: sigma_measure_b,M2: sigma_measure_b,N: sigma_measure_d,N2: sigma_measure_d] :
      ( ( ( sigma_sets_b @ M )
        = ( sigma_sets_b @ M2 ) )
     => ( ( ( sigma_sets_d @ N )
          = ( sigma_sets_d @ N2 ) )
       => ( ( sigma_measurable_b_d @ M @ N )
          = ( sigma_measurable_b_d @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_149_measurable__cong__sets,axiom,
    ! [M: sigma_measure_b,M2: sigma_measure_b,N: sigma_measure_b,N2: sigma_measure_b] :
      ( ( ( sigma_sets_b @ M )
        = ( sigma_sets_b @ M2 ) )
     => ( ( ( sigma_sets_b @ N )
          = ( sigma_sets_b @ N2 ) )
       => ( ( sigma_measurable_b_b @ M @ N )
          = ( sigma_measurable_b_b @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_150_measurable__cong__sets,axiom,
    ! [M: sigma_measure_d,M2: sigma_measure_d,N: sigma_measure_b,N2: sigma_measure_b] :
      ( ( ( sigma_sets_d @ M )
        = ( sigma_sets_d @ M2 ) )
     => ( ( ( sigma_sets_b @ N )
          = ( sigma_sets_b @ N2 ) )
       => ( ( sigma_measurable_d_b @ M @ N )
          = ( sigma_measurable_d_b @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_151_measurable__cong__sets,axiom,
    ! [M: sigma_measure_d,M2: sigma_measure_d,N: sigma_measure_d,N2: sigma_measure_d] :
      ( ( ( sigma_sets_d @ M )
        = ( sigma_sets_d @ M2 ) )
     => ( ( ( sigma_sets_d @ N )
          = ( sigma_sets_d @ N2 ) )
       => ( ( sigma_measurable_d_d @ M @ N )
          = ( sigma_measurable_d_d @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_152_measurable__cong__simp,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b,M2: sigma_measure_d,N2: sigma_measure_d,F: b > d,G: b > d] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: b] :
              ( ( member_b @ W @ ( sigma_space_b @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ M2 ) )
            = ( member_b_d @ G @ ( sigma_measurable_b_d @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_153_measurable__cong__simp,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b,M2: sigma_measure_b,N2: sigma_measure_b,F: b > b,G: b > b] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: b] :
              ( ( member_b @ W @ ( sigma_space_b @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ M2 ) )
            = ( member_b_b @ G @ ( sigma_measurable_b_b @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_154_measurable__cong__simp,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d,M2: sigma_measure_b,N2: sigma_measure_b,F: d > b,G: d > b] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: d] :
              ( ( member_d @ W @ ( sigma_space_d @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ M2 ) )
            = ( member_d_b @ G @ ( sigma_measurable_d_b @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_155_measurable__cong__simp,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d,M2: sigma_measure_d,N2: sigma_measure_d,F: d > d,G: d > d] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: d] :
              ( ( member_d @ W @ ( sigma_space_d @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ M2 ) )
            = ( member_d_d @ G @ ( sigma_measurable_d_d @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_156_measurable__space,axiom,
    ! [F: c > c,M: sigma_measure_c,A: sigma_measure_c,X: c] :
      ( ( member_c_c @ F @ ( sigma_measurable_c_c @ M @ A ) )
     => ( ( member_c @ X @ ( sigma_space_c @ M ) )
       => ( member_c @ ( F @ X ) @ ( sigma_space_c @ A ) ) ) ) ).

% measurable_space
thf(fact_157_measurable__space,axiom,
    ! [F: c > b,M: sigma_measure_c,A: sigma_measure_b,X: c] :
      ( ( member_c_b @ F @ ( sigma_measurable_c_b @ M @ A ) )
     => ( ( member_c @ X @ ( sigma_space_c @ M ) )
       => ( member_b @ ( F @ X ) @ ( sigma_space_b @ A ) ) ) ) ).

% measurable_space
thf(fact_158_measurable__space,axiom,
    ! [F: c > d,M: sigma_measure_c,A: sigma_measure_d,X: c] :
      ( ( member_c_d @ F @ ( sigma_measurable_c_d @ M @ A ) )
     => ( ( member_c @ X @ ( sigma_space_c @ M ) )
       => ( member_d @ ( F @ X ) @ ( sigma_space_d @ A ) ) ) ) ).

% measurable_space
thf(fact_159_measurable__space,axiom,
    ! [F: b > c,M: sigma_measure_b,A: sigma_measure_c,X: b] :
      ( ( member_b_c @ F @ ( sigma_measurable_b_c @ M @ A ) )
     => ( ( member_b @ X @ ( sigma_space_b @ M ) )
       => ( member_c @ ( F @ X ) @ ( sigma_space_c @ A ) ) ) ) ).

% measurable_space
thf(fact_160_measurable__space,axiom,
    ! [F: d > c,M: sigma_measure_d,A: sigma_measure_c,X: d] :
      ( ( member_d_c @ F @ ( sigma_measurable_d_c @ M @ A ) )
     => ( ( member_d @ X @ ( sigma_space_d @ M ) )
       => ( member_c @ ( F @ X ) @ ( sigma_space_c @ A ) ) ) ) ).

% measurable_space
thf(fact_161_measurable__space,axiom,
    ! [F: b > d,M: sigma_measure_b,A: sigma_measure_d,X: b] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ A ) )
     => ( ( member_b @ X @ ( sigma_space_b @ M ) )
       => ( member_d @ ( F @ X ) @ ( sigma_space_d @ A ) ) ) ) ).

% measurable_space
thf(fact_162_measurable__space,axiom,
    ! [F: b > b,M: sigma_measure_b,A: sigma_measure_b,X: b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ A ) )
     => ( ( member_b @ X @ ( sigma_space_b @ M ) )
       => ( member_b @ ( F @ X ) @ ( sigma_space_b @ A ) ) ) ) ).

% measurable_space
thf(fact_163_measurable__space,axiom,
    ! [F: d > b,M: sigma_measure_d,A: sigma_measure_b,X: d] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ A ) )
     => ( ( member_d @ X @ ( sigma_space_d @ M ) )
       => ( member_b @ ( F @ X ) @ ( sigma_space_b @ A ) ) ) ) ).

% measurable_space
thf(fact_164_measurable__space,axiom,
    ! [F: d > d,M: sigma_measure_d,A: sigma_measure_d,X: d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ A ) )
     => ( ( member_d @ X @ ( sigma_space_d @ M ) )
       => ( member_d @ ( F @ X ) @ ( sigma_space_d @ A ) ) ) ) ).

% measurable_space
thf(fact_165_measurable__space,axiom,
    ! [F: set_d > c,M: sigma_measure_set_d,A: sigma_measure_c,X: set_d] :
      ( ( member_set_d_c @ F @ ( sigma_1994698223397181103et_d_c @ M @ A ) )
     => ( ( member_set_d @ X @ ( sigma_space_set_d @ M ) )
       => ( member_c @ ( F @ X ) @ ( sigma_space_c @ A ) ) ) ) ).

% measurable_space
thf(fact_166_measurable__cong,axiom,
    ! [M: sigma_measure_b,F: b > d,G: b > d,M2: sigma_measure_d] :
      ( ! [W: b] :
          ( ( member_b @ W @ ( sigma_space_b @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ M2 ) )
        = ( member_b_d @ G @ ( sigma_measurable_b_d @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_167_measurable__cong,axiom,
    ! [M: sigma_measure_b,F: b > b,G: b > b,M2: sigma_measure_b] :
      ( ! [W: b] :
          ( ( member_b @ W @ ( sigma_space_b @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ M2 ) )
        = ( member_b_b @ G @ ( sigma_measurable_b_b @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_168_measurable__cong,axiom,
    ! [M: sigma_measure_d,F: d > b,G: d > b,M2: sigma_measure_b] :
      ( ! [W: d] :
          ( ( member_d @ W @ ( sigma_space_d @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ M2 ) )
        = ( member_d_b @ G @ ( sigma_measurable_d_b @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_169_measurable__cong,axiom,
    ! [M: sigma_measure_d,F: d > d,G: d > d,M2: sigma_measure_d] :
      ( ! [W: d] :
          ( ( member_d @ W @ ( sigma_space_d @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ M2 ) )
        = ( member_d_d @ G @ ( sigma_measurable_d_d @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_170_measurable__ident__sets,axiom,
    ! [M: sigma_measure_b,M2: sigma_measure_b] :
      ( ( ( sigma_sets_b @ M )
        = ( sigma_sets_b @ M2 ) )
     => ( member_b_b
        @ ^ [X2: b] : X2
        @ ( sigma_measurable_b_b @ M @ M2 ) ) ) ).

% measurable_ident_sets
thf(fact_171_measurable__ident__sets,axiom,
    ! [M: sigma_measure_d,M2: sigma_measure_d] :
      ( ( ( sigma_sets_d @ M )
        = ( sigma_sets_d @ M2 ) )
     => ( member_d_d
        @ ^ [X2: d] : X2
        @ ( sigma_measurable_d_d @ M @ M2 ) ) ) ).

% measurable_ident_sets
thf(fact_172_measurable__const,axiom,
    ! [C2: d,M2: sigma_measure_d,M: sigma_measure_b] :
      ( ( member_d @ C2 @ ( sigma_space_d @ M2 ) )
     => ( member_b_d
        @ ^ [X2: b] : C2
        @ ( sigma_measurable_b_d @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_173_measurable__const,axiom,
    ! [C2: b,M2: sigma_measure_b,M: sigma_measure_b] :
      ( ( member_b @ C2 @ ( sigma_space_b @ M2 ) )
     => ( member_b_b
        @ ^ [X2: b] : C2
        @ ( sigma_measurable_b_b @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_174_measurable__const,axiom,
    ! [C2: b,M2: sigma_measure_b,M: sigma_measure_d] :
      ( ( member_b @ C2 @ ( sigma_space_b @ M2 ) )
     => ( member_d_b
        @ ^ [X2: d] : C2
        @ ( sigma_measurable_d_b @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_175_measurable__const,axiom,
    ! [C2: d,M2: sigma_measure_d,M: sigma_measure_d] :
      ( ( member_d @ C2 @ ( sigma_space_d @ M2 ) )
     => ( member_d_d
        @ ^ [X2: d] : C2
        @ ( sigma_measurable_d_d @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_176_Collect__mono__iff,axiom,
    ! [P: set_b > $o,Q: set_b > $o] :
      ( ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) )
      = ( ! [X2: set_b] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_177_Collect__mono__iff,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
      = ( ! [X2: b] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_178_set__eq__subset,axiom,
    ( ( ^ [Y4: set_set_b,Z2: set_set_b] : ( Y4 = Z2 ) )
    = ( ^ [A3: set_set_b,B3: set_set_b] :
          ( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
          & ( ord_le3795704787696855135_set_b @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_179_set__eq__subset,axiom,
    ( ( ^ [Y4: set_b,Z2: set_b] : ( Y4 = Z2 ) )
    = ( ^ [A3: set_b,B3: set_b] :
          ( ( ord_less_eq_set_b @ A3 @ B3 )
          & ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_180_subset__trans,axiom,
    ! [A: set_set_b,B: set_set_b,C: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ( ord_le3795704787696855135_set_b @ B @ C )
       => ( ord_le3795704787696855135_set_b @ A @ C ) ) ) ).

% subset_trans
thf(fact_181_subset__trans,axiom,
    ! [A: set_b,B: set_b,C: set_b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( ord_less_eq_set_b @ B @ C )
       => ( ord_less_eq_set_b @ A @ C ) ) ) ).

% subset_trans
thf(fact_182_Collect__mono,axiom,
    ! [P: set_b > $o,Q: set_b > $o] :
      ( ! [X3: set_b] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) ) ) ).

% Collect_mono
thf(fact_183_Collect__mono,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ! [X3: b] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).

% Collect_mono
thf(fact_184_subset__refl,axiom,
    ! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ A @ A ) ).

% subset_refl
thf(fact_185_subset__refl,axiom,
    ! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).

% subset_refl
thf(fact_186_subset__iff,axiom,
    ( ord_le3937772867789547105_set_d
    = ( ^ [A3: set_set_d,B3: set_set_d] :
        ! [T: set_d] :
          ( ( member_set_d @ T @ A3 )
         => ( member_set_d @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_187_subset__iff,axiom,
    ( ord_less_eq_set_c
    = ( ^ [A3: set_c,B3: set_c] :
        ! [T: c] :
          ( ( member_c @ T @ A3 )
         => ( member_c @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_188_subset__iff,axiom,
    ( ord_less_eq_set_b_d
    = ( ^ [A3: set_b_d,B3: set_b_d] :
        ! [T: b > d] :
          ( ( member_b_d @ T @ A3 )
         => ( member_b_d @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_189_subset__iff,axiom,
    ( ord_less_eq_set_b_b
    = ( ^ [A3: set_b_b,B3: set_b_b] :
        ! [T: b > b] :
          ( ( member_b_b @ T @ A3 )
         => ( member_b_b @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_190_subset__iff,axiom,
    ( ord_less_eq_set_d_b
    = ( ^ [A3: set_d_b,B3: set_d_b] :
        ! [T: d > b] :
          ( ( member_d_b @ T @ A3 )
         => ( member_d_b @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_191_subset__iff,axiom,
    ( ord_less_eq_set_d_d
    = ( ^ [A3: set_d_d,B3: set_d_d] :
        ! [T: d > d] :
          ( ( member_d_d @ T @ A3 )
         => ( member_d_d @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_192_subset__iff,axiom,
    ( ord_le3795704787696855135_set_b
    = ( ^ [A3: set_set_b,B3: set_set_b] :
        ! [T: set_b] :
          ( ( member_set_b @ T @ A3 )
         => ( member_set_b @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_193_subset__iff,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A3: set_b,B3: set_b] :
        ! [T: b] :
          ( ( member_b @ T @ A3 )
         => ( member_b @ T @ B3 ) ) ) ) ).

% subset_iff
thf(fact_194_mem__Collect__eq,axiom,
    ! [A2: set_d,P: set_d > $o] :
      ( ( member_set_d @ A2 @ ( collect_set_d @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_195_mem__Collect__eq,axiom,
    ! [A2: c,P: c > $o] :
      ( ( member_c @ A2 @ ( collect_c @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_196_mem__Collect__eq,axiom,
    ! [A2: b > d,P: ( b > d ) > $o] :
      ( ( member_b_d @ A2 @ ( collect_b_d @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_197_mem__Collect__eq,axiom,
    ! [A2: b > b,P: ( b > b ) > $o] :
      ( ( member_b_b @ A2 @ ( collect_b_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_198_mem__Collect__eq,axiom,
    ! [A2: d > b,P: ( d > b ) > $o] :
      ( ( member_d_b @ A2 @ ( collect_d_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_199_mem__Collect__eq,axiom,
    ! [A2: d > d,P: ( d > d ) > $o] :
      ( ( member_d_d @ A2 @ ( collect_d_d @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_200_mem__Collect__eq,axiom,
    ! [A2: set_b,P: set_b > $o] :
      ( ( member_set_b @ A2 @ ( collect_set_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_201_Collect__mem__eq,axiom,
    ! [A: set_set_d] :
      ( ( collect_set_d
        @ ^ [X2: set_d] : ( member_set_d @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_202_Collect__mem__eq,axiom,
    ! [A: set_c] :
      ( ( collect_c
        @ ^ [X2: c] : ( member_c @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_203_Collect__mem__eq,axiom,
    ! [A: set_b_d] :
      ( ( collect_b_d
        @ ^ [X2: b > d] : ( member_b_d @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_204_Collect__mem__eq,axiom,
    ! [A: set_b_b] :
      ( ( collect_b_b
        @ ^ [X2: b > b] : ( member_b_b @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_205_Collect__mem__eq,axiom,
    ! [A: set_d_b] :
      ( ( collect_d_b
        @ ^ [X2: d > b] : ( member_d_b @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_206_Collect__mem__eq,axiom,
    ! [A: set_d_d] :
      ( ( collect_d_d
        @ ^ [X2: d > d] : ( member_d_d @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_207_Collect__mem__eq,axiom,
    ! [A: set_set_b] :
      ( ( collect_set_b
        @ ^ [X2: set_b] : ( member_set_b @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_208_Collect__cong,axiom,
    ! [P: set_b > $o,Q: set_b > $o] :
      ( ! [X3: set_b] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_b @ P )
        = ( collect_set_b @ Q ) ) ) ).

% Collect_cong
thf(fact_209_equalityD2,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( A = B )
     => ( ord_le3795704787696855135_set_b @ B @ A ) ) ).

% equalityD2
thf(fact_210_equalityD2,axiom,
    ! [A: set_b,B: set_b] :
      ( ( A = B )
     => ( ord_less_eq_set_b @ B @ A ) ) ).

% equalityD2
thf(fact_211_equalityD1,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( A = B )
     => ( ord_le3795704787696855135_set_b @ A @ B ) ) ).

% equalityD1
thf(fact_212_equalityD1,axiom,
    ! [A: set_b,B: set_b] :
      ( ( A = B )
     => ( ord_less_eq_set_b @ A @ B ) ) ).

% equalityD1
thf(fact_213_subset__eq,axiom,
    ( ord_le3937772867789547105_set_d
    = ( ^ [A3: set_set_d,B3: set_set_d] :
        ! [X2: set_d] :
          ( ( member_set_d @ X2 @ A3 )
         => ( member_set_d @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_214_subset__eq,axiom,
    ( ord_less_eq_set_c
    = ( ^ [A3: set_c,B3: set_c] :
        ! [X2: c] :
          ( ( member_c @ X2 @ A3 )
         => ( member_c @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_215_subset__eq,axiom,
    ( ord_less_eq_set_b_d
    = ( ^ [A3: set_b_d,B3: set_b_d] :
        ! [X2: b > d] :
          ( ( member_b_d @ X2 @ A3 )
         => ( member_b_d @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_216_subset__eq,axiom,
    ( ord_less_eq_set_b_b
    = ( ^ [A3: set_b_b,B3: set_b_b] :
        ! [X2: b > b] :
          ( ( member_b_b @ X2 @ A3 )
         => ( member_b_b @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_217_subset__eq,axiom,
    ( ord_less_eq_set_d_b
    = ( ^ [A3: set_d_b,B3: set_d_b] :
        ! [X2: d > b] :
          ( ( member_d_b @ X2 @ A3 )
         => ( member_d_b @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_218_subset__eq,axiom,
    ( ord_less_eq_set_d_d
    = ( ^ [A3: set_d_d,B3: set_d_d] :
        ! [X2: d > d] :
          ( ( member_d_d @ X2 @ A3 )
         => ( member_d_d @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_219_subset__eq,axiom,
    ( ord_le3795704787696855135_set_b
    = ( ^ [A3: set_set_b,B3: set_set_b] :
        ! [X2: set_b] :
          ( ( member_set_b @ X2 @ A3 )
         => ( member_set_b @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_220_subset__eq,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A3: set_b,B3: set_b] :
        ! [X2: b] :
          ( ( member_b @ X2 @ A3 )
         => ( member_b @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_221_equalityE,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( A = B )
     => ~ ( ( ord_le3795704787696855135_set_b @ A @ B )
         => ~ ( ord_le3795704787696855135_set_b @ B @ A ) ) ) ).

% equalityE
thf(fact_222_equalityE,axiom,
    ! [A: set_b,B: set_b] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_b @ A @ B )
         => ~ ( ord_less_eq_set_b @ B @ A ) ) ) ).

% equalityE
thf(fact_223_subsetD,axiom,
    ! [A: set_set_d,B: set_set_d,C2: set_d] :
      ( ( ord_le3937772867789547105_set_d @ A @ B )
     => ( ( member_set_d @ C2 @ A )
       => ( member_set_d @ C2 @ B ) ) ) ).

% subsetD
thf(fact_224_subsetD,axiom,
    ! [A: set_c,B: set_c,C2: c] :
      ( ( ord_less_eq_set_c @ A @ B )
     => ( ( member_c @ C2 @ A )
       => ( member_c @ C2 @ B ) ) ) ).

% subsetD
thf(fact_225_subsetD,axiom,
    ! [A: set_b_d,B: set_b_d,C2: b > d] :
      ( ( ord_less_eq_set_b_d @ A @ B )
     => ( ( member_b_d @ C2 @ A )
       => ( member_b_d @ C2 @ B ) ) ) ).

% subsetD
thf(fact_226_subsetD,axiom,
    ! [A: set_b_b,B: set_b_b,C2: b > b] :
      ( ( ord_less_eq_set_b_b @ A @ B )
     => ( ( member_b_b @ C2 @ A )
       => ( member_b_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_227_subsetD,axiom,
    ! [A: set_d_b,B: set_d_b,C2: d > b] :
      ( ( ord_less_eq_set_d_b @ A @ B )
     => ( ( member_d_b @ C2 @ A )
       => ( member_d_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_228_subsetD,axiom,
    ! [A: set_d_d,B: set_d_d,C2: d > d] :
      ( ( ord_less_eq_set_d_d @ A @ B )
     => ( ( member_d_d @ C2 @ A )
       => ( member_d_d @ C2 @ B ) ) ) ).

% subsetD
thf(fact_229_subsetD,axiom,
    ! [A: set_set_b,B: set_set_b,C2: set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ( member_set_b @ C2 @ A )
       => ( member_set_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_230_subsetD,axiom,
    ! [A: set_b,B: set_b,C2: b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( member_b @ C2 @ A )
       => ( member_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_231_in__mono,axiom,
    ! [A: set_set_d,B: set_set_d,X: set_d] :
      ( ( ord_le3937772867789547105_set_d @ A @ B )
     => ( ( member_set_d @ X @ A )
       => ( member_set_d @ X @ B ) ) ) ).

% in_mono
thf(fact_232_in__mono,axiom,
    ! [A: set_c,B: set_c,X: c] :
      ( ( ord_less_eq_set_c @ A @ B )
     => ( ( member_c @ X @ A )
       => ( member_c @ X @ B ) ) ) ).

% in_mono
thf(fact_233_in__mono,axiom,
    ! [A: set_b_d,B: set_b_d,X: b > d] :
      ( ( ord_less_eq_set_b_d @ A @ B )
     => ( ( member_b_d @ X @ A )
       => ( member_b_d @ X @ B ) ) ) ).

% in_mono
thf(fact_234_in__mono,axiom,
    ! [A: set_b_b,B: set_b_b,X: b > b] :
      ( ( ord_less_eq_set_b_b @ A @ B )
     => ( ( member_b_b @ X @ A )
       => ( member_b_b @ X @ B ) ) ) ).

% in_mono
thf(fact_235_in__mono,axiom,
    ! [A: set_d_b,B: set_d_b,X: d > b] :
      ( ( ord_less_eq_set_d_b @ A @ B )
     => ( ( member_d_b @ X @ A )
       => ( member_d_b @ X @ B ) ) ) ).

% in_mono
thf(fact_236_in__mono,axiom,
    ! [A: set_d_d,B: set_d_d,X: d > d] :
      ( ( ord_less_eq_set_d_d @ A @ B )
     => ( ( member_d_d @ X @ A )
       => ( member_d_d @ X @ B ) ) ) ).

% in_mono
thf(fact_237_in__mono,axiom,
    ! [A: set_set_b,B: set_set_b,X: set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ( member_set_b @ X @ A )
       => ( member_set_b @ X @ B ) ) ) ).

% in_mono
thf(fact_238_in__mono,axiom,
    ! [A: set_b,B: set_b,X: b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( member_b @ X @ A )
       => ( member_b @ X @ B ) ) ) ).

% in_mono
thf(fact_239_inf__left__commute,axiom,
    ! [X: set_b,Y: set_b,Z: set_b] :
      ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
      = ( inf_inf_set_b @ Y @ ( inf_inf_set_b @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_240_inf__left__commute,axiom,
    ! [X: set_d,Y: set_d,Z: set_d] :
      ( ( inf_inf_set_d @ X @ ( inf_inf_set_d @ Y @ Z ) )
      = ( inf_inf_set_d @ Y @ ( inf_inf_set_d @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_241_inf__left__commute,axiom,
    ! [X: set_set_b,Y: set_set_b,Z: set_set_b] :
      ( ( inf_inf_set_set_b @ X @ ( inf_inf_set_set_b @ Y @ Z ) )
      = ( inf_inf_set_set_b @ Y @ ( inf_inf_set_set_b @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_242_inf_Oleft__commute,axiom,
    ! [B2: set_b,A2: set_b,C2: set_b] :
      ( ( inf_inf_set_b @ B2 @ ( inf_inf_set_b @ A2 @ C2 ) )
      = ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C2 ) ) ) ).

% inf.left_commute
thf(fact_243_inf_Oleft__commute,axiom,
    ! [B2: set_d,A2: set_d,C2: set_d] :
      ( ( inf_inf_set_d @ B2 @ ( inf_inf_set_d @ A2 @ C2 ) )
      = ( inf_inf_set_d @ A2 @ ( inf_inf_set_d @ B2 @ C2 ) ) ) ).

% inf.left_commute
thf(fact_244_inf_Oleft__commute,axiom,
    ! [B2: set_set_b,A2: set_set_b,C2: set_set_b] :
      ( ( inf_inf_set_set_b @ B2 @ ( inf_inf_set_set_b @ A2 @ C2 ) )
      = ( inf_inf_set_set_b @ A2 @ ( inf_inf_set_set_b @ B2 @ C2 ) ) ) ).

% inf.left_commute
thf(fact_245_inf__commute,axiom,
    ( inf_inf_set_b
    = ( ^ [X2: set_b,Y3: set_b] : ( inf_inf_set_b @ Y3 @ X2 ) ) ) ).

% inf_commute
thf(fact_246_inf__commute,axiom,
    ( inf_inf_set_d
    = ( ^ [X2: set_d,Y3: set_d] : ( inf_inf_set_d @ Y3 @ X2 ) ) ) ).

% inf_commute
thf(fact_247_inf__commute,axiom,
    ( inf_inf_set_set_b
    = ( ^ [X2: set_set_b,Y3: set_set_b] : ( inf_inf_set_set_b @ Y3 @ X2 ) ) ) ).

% inf_commute
thf(fact_248_inf_Ocommute,axiom,
    ( inf_inf_set_b
    = ( ^ [A4: set_b,B4: set_b] : ( inf_inf_set_b @ B4 @ A4 ) ) ) ).

% inf.commute
thf(fact_249_inf_Ocommute,axiom,
    ( inf_inf_set_d
    = ( ^ [A4: set_d,B4: set_d] : ( inf_inf_set_d @ B4 @ A4 ) ) ) ).

% inf.commute
thf(fact_250_inf_Ocommute,axiom,
    ( inf_inf_set_set_b
    = ( ^ [A4: set_set_b,B4: set_set_b] : ( inf_inf_set_set_b @ B4 @ A4 ) ) ) ).

% inf.commute
thf(fact_251_inf__assoc,axiom,
    ! [X: set_b,Y: set_b,Z: set_b] :
      ( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Z )
      = ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_252_inf__assoc,axiom,
    ! [X: set_d,Y: set_d,Z: set_d] :
      ( ( inf_inf_set_d @ ( inf_inf_set_d @ X @ Y ) @ Z )
      = ( inf_inf_set_d @ X @ ( inf_inf_set_d @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_253_inf__assoc,axiom,
    ! [X: set_set_b,Y: set_set_b,Z: set_set_b] :
      ( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ X @ Y ) @ Z )
      = ( inf_inf_set_set_b @ X @ ( inf_inf_set_set_b @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_254_inf_Oassoc,axiom,
    ! [A2: set_b,B2: set_b,C2: set_b] :
      ( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C2 )
      = ( inf_inf_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C2 ) ) ) ).

% inf.assoc
thf(fact_255_inf_Oassoc,axiom,
    ! [A2: set_d,B2: set_d,C2: set_d] :
      ( ( inf_inf_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ C2 )
      = ( inf_inf_set_d @ A2 @ ( inf_inf_set_d @ B2 @ C2 ) ) ) ).

% inf.assoc
thf(fact_256_inf_Oassoc,axiom,
    ! [A2: set_set_b,B2: set_set_b,C2: set_set_b] :
      ( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ C2 )
      = ( inf_inf_set_set_b @ A2 @ ( inf_inf_set_set_b @ B2 @ C2 ) ) ) ).

% inf.assoc
thf(fact_257_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_b
    = ( ^ [X2: set_b,Y3: set_b] : ( inf_inf_set_b @ Y3 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_258_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_d
    = ( ^ [X2: set_d,Y3: set_d] : ( inf_inf_set_d @ Y3 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_259_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_set_b
    = ( ^ [X2: set_set_b,Y3: set_set_b] : ( inf_inf_set_set_b @ Y3 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_260_inf__sup__aci_I2_J,axiom,
    ! [X: set_b,Y: set_b,Z: set_b] :
      ( ( inf_inf_set_b @ ( inf_inf_set_b @ X @ Y ) @ Z )
      = ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_261_inf__sup__aci_I2_J,axiom,
    ! [X: set_d,Y: set_d,Z: set_d] :
      ( ( inf_inf_set_d @ ( inf_inf_set_d @ X @ Y ) @ Z )
      = ( inf_inf_set_d @ X @ ( inf_inf_set_d @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_262_inf__sup__aci_I2_J,axiom,
    ! [X: set_set_b,Y: set_set_b,Z: set_set_b] :
      ( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ X @ Y ) @ Z )
      = ( inf_inf_set_set_b @ X @ ( inf_inf_set_set_b @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_263_inf__sup__aci_I3_J,axiom,
    ! [X: set_b,Y: set_b,Z: set_b] :
      ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
      = ( inf_inf_set_b @ Y @ ( inf_inf_set_b @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_264_inf__sup__aci_I3_J,axiom,
    ! [X: set_d,Y: set_d,Z: set_d] :
      ( ( inf_inf_set_d @ X @ ( inf_inf_set_d @ Y @ Z ) )
      = ( inf_inf_set_d @ Y @ ( inf_inf_set_d @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_265_inf__sup__aci_I3_J,axiom,
    ! [X: set_set_b,Y: set_set_b,Z: set_set_b] :
      ( ( inf_inf_set_set_b @ X @ ( inf_inf_set_set_b @ Y @ Z ) )
      = ( inf_inf_set_set_b @ Y @ ( inf_inf_set_set_b @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_266_inf__sup__aci_I4_J,axiom,
    ! [X: set_b,Y: set_b] :
      ( ( inf_inf_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
      = ( inf_inf_set_b @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_267_inf__sup__aci_I4_J,axiom,
    ! [X: set_d,Y: set_d] :
      ( ( inf_inf_set_d @ X @ ( inf_inf_set_d @ X @ Y ) )
      = ( inf_inf_set_d @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_268_inf__sup__aci_I4_J,axiom,
    ! [X: set_set_b,Y: set_set_b] :
      ( ( inf_inf_set_set_b @ X @ ( inf_inf_set_set_b @ X @ Y ) )
      = ( inf_inf_set_set_b @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_269_Int__left__commute,axiom,
    ! [A: set_b,B: set_b,C: set_b] :
      ( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C ) )
      = ( inf_inf_set_b @ B @ ( inf_inf_set_b @ A @ C ) ) ) ).

% Int_left_commute
thf(fact_270_Int__left__commute,axiom,
    ! [A: set_d,B: set_d,C: set_d] :
      ( ( inf_inf_set_d @ A @ ( inf_inf_set_d @ B @ C ) )
      = ( inf_inf_set_d @ B @ ( inf_inf_set_d @ A @ C ) ) ) ).

% Int_left_commute
thf(fact_271_Int__left__commute,axiom,
    ! [A: set_set_b,B: set_set_b,C: set_set_b] :
      ( ( inf_inf_set_set_b @ A @ ( inf_inf_set_set_b @ B @ C ) )
      = ( inf_inf_set_set_b @ B @ ( inf_inf_set_set_b @ A @ C ) ) ) ).

% Int_left_commute
thf(fact_272_Int__left__absorb,axiom,
    ! [A: set_b,B: set_b] :
      ( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ A @ B ) )
      = ( inf_inf_set_b @ A @ B ) ) ).

% Int_left_absorb
thf(fact_273_Int__left__absorb,axiom,
    ! [A: set_d,B: set_d] :
      ( ( inf_inf_set_d @ A @ ( inf_inf_set_d @ A @ B ) )
      = ( inf_inf_set_d @ A @ B ) ) ).

% Int_left_absorb
thf(fact_274_Int__left__absorb,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( inf_inf_set_set_b @ A @ ( inf_inf_set_set_b @ A @ B ) )
      = ( inf_inf_set_set_b @ A @ B ) ) ).

% Int_left_absorb
thf(fact_275_Int__commute,axiom,
    ( inf_inf_set_b
    = ( ^ [A3: set_b,B3: set_b] : ( inf_inf_set_b @ B3 @ A3 ) ) ) ).

% Int_commute
thf(fact_276_Int__commute,axiom,
    ( inf_inf_set_d
    = ( ^ [A3: set_d,B3: set_d] : ( inf_inf_set_d @ B3 @ A3 ) ) ) ).

% Int_commute
thf(fact_277_Int__commute,axiom,
    ( inf_inf_set_set_b
    = ( ^ [A3: set_set_b,B3: set_set_b] : ( inf_inf_set_set_b @ B3 @ A3 ) ) ) ).

% Int_commute
thf(fact_278_Int__absorb,axiom,
    ! [A: set_b] :
      ( ( inf_inf_set_b @ A @ A )
      = A ) ).

% Int_absorb
thf(fact_279_Int__absorb,axiom,
    ! [A: set_d] :
      ( ( inf_inf_set_d @ A @ A )
      = A ) ).

% Int_absorb
thf(fact_280_Int__absorb,axiom,
    ! [A: set_set_b] :
      ( ( inf_inf_set_set_b @ A @ A )
      = A ) ).

% Int_absorb
thf(fact_281_Int__assoc,axiom,
    ! [A: set_b,B: set_b,C: set_b] :
      ( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B ) @ C )
      = ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C ) ) ) ).

% Int_assoc
thf(fact_282_Int__assoc,axiom,
    ! [A: set_d,B: set_d,C: set_d] :
      ( ( inf_inf_set_d @ ( inf_inf_set_d @ A @ B ) @ C )
      = ( inf_inf_set_d @ A @ ( inf_inf_set_d @ B @ C ) ) ) ).

% Int_assoc
thf(fact_283_Int__assoc,axiom,
    ! [A: set_set_b,B: set_set_b,C: set_set_b] :
      ( ( inf_inf_set_set_b @ ( inf_inf_set_set_b @ A @ B ) @ C )
      = ( inf_inf_set_set_b @ A @ ( inf_inf_set_set_b @ B @ C ) ) ) ).

% Int_assoc
thf(fact_284_IntD2,axiom,
    ! [C2: set_d,A: set_set_d,B: set_set_d] :
      ( ( member_set_d @ C2 @ ( inf_inf_set_set_d @ A @ B ) )
     => ( member_set_d @ C2 @ B ) ) ).

% IntD2
thf(fact_285_IntD2,axiom,
    ! [C2: c,A: set_c,B: set_c] :
      ( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B ) )
     => ( member_c @ C2 @ B ) ) ).

% IntD2
thf(fact_286_IntD2,axiom,
    ! [C2: b > d,A: set_b_d,B: set_b_d] :
      ( ( member_b_d @ C2 @ ( inf_inf_set_b_d @ A @ B ) )
     => ( member_b_d @ C2 @ B ) ) ).

% IntD2
thf(fact_287_IntD2,axiom,
    ! [C2: b > b,A: set_b_b,B: set_b_b] :
      ( ( member_b_b @ C2 @ ( inf_inf_set_b_b @ A @ B ) )
     => ( member_b_b @ C2 @ B ) ) ).

% IntD2
thf(fact_288_IntD2,axiom,
    ! [C2: d > b,A: set_d_b,B: set_d_b] :
      ( ( member_d_b @ C2 @ ( inf_inf_set_d_b @ A @ B ) )
     => ( member_d_b @ C2 @ B ) ) ).

% IntD2
thf(fact_289_IntD2,axiom,
    ! [C2: d > d,A: set_d_d,B: set_d_d] :
      ( ( member_d_d @ C2 @ ( inf_inf_set_d_d @ A @ B ) )
     => ( member_d_d @ C2 @ B ) ) ).

% IntD2
thf(fact_290_IntD2,axiom,
    ! [C2: b,A: set_b,B: set_b] :
      ( ( member_b @ C2 @ ( inf_inf_set_b @ A @ B ) )
     => ( member_b @ C2 @ B ) ) ).

% IntD2
thf(fact_291_IntD2,axiom,
    ! [C2: d,A: set_d,B: set_d] :
      ( ( member_d @ C2 @ ( inf_inf_set_d @ A @ B ) )
     => ( member_d @ C2 @ B ) ) ).

% IntD2
thf(fact_292_IntD2,axiom,
    ! [C2: set_b,A: set_set_b,B: set_set_b] :
      ( ( member_set_b @ C2 @ ( inf_inf_set_set_b @ A @ B ) )
     => ( member_set_b @ C2 @ B ) ) ).

% IntD2
thf(fact_293_IntD1,axiom,
    ! [C2: set_d,A: set_set_d,B: set_set_d] :
      ( ( member_set_d @ C2 @ ( inf_inf_set_set_d @ A @ B ) )
     => ( member_set_d @ C2 @ A ) ) ).

% IntD1
thf(fact_294_IntD1,axiom,
    ! [C2: c,A: set_c,B: set_c] :
      ( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B ) )
     => ( member_c @ C2 @ A ) ) ).

% IntD1
thf(fact_295_IntD1,axiom,
    ! [C2: b > d,A: set_b_d,B: set_b_d] :
      ( ( member_b_d @ C2 @ ( inf_inf_set_b_d @ A @ B ) )
     => ( member_b_d @ C2 @ A ) ) ).

% IntD1
thf(fact_296_IntD1,axiom,
    ! [C2: b > b,A: set_b_b,B: set_b_b] :
      ( ( member_b_b @ C2 @ ( inf_inf_set_b_b @ A @ B ) )
     => ( member_b_b @ C2 @ A ) ) ).

% IntD1
thf(fact_297_IntD1,axiom,
    ! [C2: d > b,A: set_d_b,B: set_d_b] :
      ( ( member_d_b @ C2 @ ( inf_inf_set_d_b @ A @ B ) )
     => ( member_d_b @ C2 @ A ) ) ).

% IntD1
thf(fact_298_IntD1,axiom,
    ! [C2: d > d,A: set_d_d,B: set_d_d] :
      ( ( member_d_d @ C2 @ ( inf_inf_set_d_d @ A @ B ) )
     => ( member_d_d @ C2 @ A ) ) ).

% IntD1
thf(fact_299_IntD1,axiom,
    ! [C2: b,A: set_b,B: set_b] :
      ( ( member_b @ C2 @ ( inf_inf_set_b @ A @ B ) )
     => ( member_b @ C2 @ A ) ) ).

% IntD1
thf(fact_300_IntD1,axiom,
    ! [C2: d,A: set_d,B: set_d] :
      ( ( member_d @ C2 @ ( inf_inf_set_d @ A @ B ) )
     => ( member_d @ C2 @ A ) ) ).

% IntD1
thf(fact_301_IntD1,axiom,
    ! [C2: set_b,A: set_set_b,B: set_set_b] :
      ( ( member_set_b @ C2 @ ( inf_inf_set_set_b @ A @ B ) )
     => ( member_set_b @ C2 @ A ) ) ).

% IntD1
thf(fact_302_IntE,axiom,
    ! [C2: set_d,A: set_set_d,B: set_set_d] :
      ( ( member_set_d @ C2 @ ( inf_inf_set_set_d @ A @ B ) )
     => ~ ( ( member_set_d @ C2 @ A )
         => ~ ( member_set_d @ C2 @ B ) ) ) ).

% IntE
thf(fact_303_IntE,axiom,
    ! [C2: c,A: set_c,B: set_c] :
      ( ( member_c @ C2 @ ( inf_inf_set_c @ A @ B ) )
     => ~ ( ( member_c @ C2 @ A )
         => ~ ( member_c @ C2 @ B ) ) ) ).

% IntE
thf(fact_304_IntE,axiom,
    ! [C2: b > d,A: set_b_d,B: set_b_d] :
      ( ( member_b_d @ C2 @ ( inf_inf_set_b_d @ A @ B ) )
     => ~ ( ( member_b_d @ C2 @ A )
         => ~ ( member_b_d @ C2 @ B ) ) ) ).

% IntE
thf(fact_305_IntE,axiom,
    ! [C2: b > b,A: set_b_b,B: set_b_b] :
      ( ( member_b_b @ C2 @ ( inf_inf_set_b_b @ A @ B ) )
     => ~ ( ( member_b_b @ C2 @ A )
         => ~ ( member_b_b @ C2 @ B ) ) ) ).

% IntE
thf(fact_306_IntE,axiom,
    ! [C2: d > b,A: set_d_b,B: set_d_b] :
      ( ( member_d_b @ C2 @ ( inf_inf_set_d_b @ A @ B ) )
     => ~ ( ( member_d_b @ C2 @ A )
         => ~ ( member_d_b @ C2 @ B ) ) ) ).

% IntE
thf(fact_307_IntE,axiom,
    ! [C2: d > d,A: set_d_d,B: set_d_d] :
      ( ( member_d_d @ C2 @ ( inf_inf_set_d_d @ A @ B ) )
     => ~ ( ( member_d_d @ C2 @ A )
         => ~ ( member_d_d @ C2 @ B ) ) ) ).

% IntE
thf(fact_308_IntE,axiom,
    ! [C2: b,A: set_b,B: set_b] :
      ( ( member_b @ C2 @ ( inf_inf_set_b @ A @ B ) )
     => ~ ( ( member_b @ C2 @ A )
         => ~ ( member_b @ C2 @ B ) ) ) ).

% IntE
thf(fact_309_IntE,axiom,
    ! [C2: d,A: set_d,B: set_d] :
      ( ( member_d @ C2 @ ( inf_inf_set_d @ A @ B ) )
     => ~ ( ( member_d @ C2 @ A )
         => ~ ( member_d @ C2 @ B ) ) ) ).

% IntE
thf(fact_310_IntE,axiom,
    ! [C2: set_b,A: set_set_b,B: set_set_b] :
      ( ( member_set_b @ C2 @ ( inf_inf_set_set_b @ A @ B ) )
     => ~ ( ( member_set_b @ C2 @ A )
         => ~ ( member_set_b @ C2 @ B ) ) ) ).

% IntE
thf(fact_311_vimage__Collect,axiom,
    ! [P: d > $o,F: b > d,Q: b > $o] :
      ( ! [X3: b] :
          ( ( P @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage_b_d @ F @ ( collect_d @ P ) )
        = ( collect_b @ Q ) ) ) ).

% vimage_Collect
thf(fact_312_vimage__Collect,axiom,
    ! [P: b > $o,F: b > b,Q: b > $o] :
      ( ! [X3: b] :
          ( ( P @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage_b_b @ F @ ( collect_b @ P ) )
        = ( collect_b @ Q ) ) ) ).

% vimage_Collect
thf(fact_313_vimage__Collect,axiom,
    ! [P: b > $o,F: d > b,Q: d > $o] :
      ( ! [X3: d] :
          ( ( P @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage_d_b @ F @ ( collect_b @ P ) )
        = ( collect_d @ Q ) ) ) ).

% vimage_Collect
thf(fact_314_vimage__Collect,axiom,
    ! [P: d > $o,F: d > d,Q: d > $o] :
      ( ! [X3: d] :
          ( ( P @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage_d_d @ F @ ( collect_d @ P ) )
        = ( collect_d @ Q ) ) ) ).

% vimage_Collect
thf(fact_315_vimage__Collect,axiom,
    ! [P: set_b > $o,F: set_b > set_b,Q: set_b > $o] :
      ( ! [X3: set_b] :
          ( ( P @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage_set_b_set_b @ F @ ( collect_set_b @ P ) )
        = ( collect_set_b @ Q ) ) ) ).

% vimage_Collect
thf(fact_316_vimageI2,axiom,
    ! [F: c > c,A2: c,A: set_c] :
      ( ( member_c @ ( F @ A2 ) @ A )
     => ( member_c @ A2 @ ( vimage_c_c @ F @ A ) ) ) ).

% vimageI2
thf(fact_317_vimageI2,axiom,
    ! [F: b > d,A2: b,A: set_d] :
      ( ( member_d @ ( F @ A2 ) @ A )
     => ( member_b @ A2 @ ( vimage_b_d @ F @ A ) ) ) ).

% vimageI2
thf(fact_318_vimageI2,axiom,
    ! [F: b > b,A2: b,A: set_b] :
      ( ( member_b @ ( F @ A2 ) @ A )
     => ( member_b @ A2 @ ( vimage_b_b @ F @ A ) ) ) ).

% vimageI2
thf(fact_319_vimageI2,axiom,
    ! [F: d > b,A2: d,A: set_b] :
      ( ( member_b @ ( F @ A2 ) @ A )
     => ( member_d @ A2 @ ( vimage_d_b @ F @ A ) ) ) ).

% vimageI2
thf(fact_320_vimageI2,axiom,
    ! [F: d > d,A2: d,A: set_d] :
      ( ( member_d @ ( F @ A2 ) @ A )
     => ( member_d @ A2 @ ( vimage_d_d @ F @ A ) ) ) ).

% vimageI2
thf(fact_321_vimageI2,axiom,
    ! [F: c > set_d,A2: c,A: set_set_d] :
      ( ( member_set_d @ ( F @ A2 ) @ A )
     => ( member_c @ A2 @ ( vimage_c_set_d @ F @ A ) ) ) ).

% vimageI2
thf(fact_322_vimageI2,axiom,
    ! [F: set_d > c,A2: set_d,A: set_c] :
      ( ( member_c @ ( F @ A2 ) @ A )
     => ( member_set_d @ A2 @ ( vimage_set_d_c @ F @ A ) ) ) ).

% vimageI2
thf(fact_323_vimageI2,axiom,
    ! [F: set_b > c,A2: set_b,A: set_c] :
      ( ( member_c @ ( F @ A2 ) @ A )
     => ( member_set_b @ A2 @ ( vimage_set_b_c @ F @ A ) ) ) ).

% vimageI2
thf(fact_324_vimageI2,axiom,
    ! [F: c > set_b,A2: c,A: set_set_b] :
      ( ( member_set_b @ ( F @ A2 ) @ A )
     => ( member_c @ A2 @ ( vimage_c_set_b @ F @ A ) ) ) ).

% vimageI2
thf(fact_325_vimageI2,axiom,
    ! [F: set_d > set_d,A2: set_d,A: set_set_d] :
      ( ( member_set_d @ ( F @ A2 ) @ A )
     => ( member_set_d @ A2 @ ( vimage_set_d_set_d @ F @ A ) ) ) ).

% vimageI2
thf(fact_326_vimageE,axiom,
    ! [A2: c,F: c > c,B: set_c] :
      ( ( member_c @ A2 @ ( vimage_c_c @ F @ B ) )
     => ( member_c @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_327_vimageE,axiom,
    ! [A2: b,F: b > d,B: set_d] :
      ( ( member_b @ A2 @ ( vimage_b_d @ F @ B ) )
     => ( member_d @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_328_vimageE,axiom,
    ! [A2: b,F: b > b,B: set_b] :
      ( ( member_b @ A2 @ ( vimage_b_b @ F @ B ) )
     => ( member_b @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_329_vimageE,axiom,
    ! [A2: d,F: d > b,B: set_b] :
      ( ( member_d @ A2 @ ( vimage_d_b @ F @ B ) )
     => ( member_b @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_330_vimageE,axiom,
    ! [A2: d,F: d > d,B: set_d] :
      ( ( member_d @ A2 @ ( vimage_d_d @ F @ B ) )
     => ( member_d @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_331_vimageE,axiom,
    ! [A2: set_d,F: set_d > c,B: set_c] :
      ( ( member_set_d @ A2 @ ( vimage_set_d_c @ F @ B ) )
     => ( member_c @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_332_vimageE,axiom,
    ! [A2: c,F: c > set_d,B: set_set_d] :
      ( ( member_c @ A2 @ ( vimage_c_set_d @ F @ B ) )
     => ( member_set_d @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_333_vimageE,axiom,
    ! [A2: c,F: c > set_b,B: set_set_b] :
      ( ( member_c @ A2 @ ( vimage_c_set_b @ F @ B ) )
     => ( member_set_b @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_334_vimageE,axiom,
    ! [A2: set_b,F: set_b > c,B: set_c] :
      ( ( member_set_b @ A2 @ ( vimage_set_b_c @ F @ B ) )
     => ( member_c @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_335_vimageE,axiom,
    ! [A2: set_d,F: set_d > set_d,B: set_set_d] :
      ( ( member_set_d @ A2 @ ( vimage_set_d_set_d @ F @ B ) )
     => ( member_set_d @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_336_vimageD,axiom,
    ! [A2: c,F: c > c,A: set_c] :
      ( ( member_c @ A2 @ ( vimage_c_c @ F @ A ) )
     => ( member_c @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_337_vimageD,axiom,
    ! [A2: b,F: b > d,A: set_d] :
      ( ( member_b @ A2 @ ( vimage_b_d @ F @ A ) )
     => ( member_d @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_338_vimageD,axiom,
    ! [A2: b,F: b > b,A: set_b] :
      ( ( member_b @ A2 @ ( vimage_b_b @ F @ A ) )
     => ( member_b @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_339_vimageD,axiom,
    ! [A2: d,F: d > b,A: set_b] :
      ( ( member_d @ A2 @ ( vimage_d_b @ F @ A ) )
     => ( member_b @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_340_vimageD,axiom,
    ! [A2: d,F: d > d,A: set_d] :
      ( ( member_d @ A2 @ ( vimage_d_d @ F @ A ) )
     => ( member_d @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_341_vimageD,axiom,
    ! [A2: set_d,F: set_d > c,A: set_c] :
      ( ( member_set_d @ A2 @ ( vimage_set_d_c @ F @ A ) )
     => ( member_c @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_342_vimageD,axiom,
    ! [A2: c,F: c > set_d,A: set_set_d] :
      ( ( member_c @ A2 @ ( vimage_c_set_d @ F @ A ) )
     => ( member_set_d @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_343_vimageD,axiom,
    ! [A2: c,F: c > set_b,A: set_set_b] :
      ( ( member_c @ A2 @ ( vimage_c_set_b @ F @ A ) )
     => ( member_set_b @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_344_vimageD,axiom,
    ! [A2: set_b,F: set_b > c,A: set_c] :
      ( ( member_set_b @ A2 @ ( vimage_set_b_c @ F @ A ) )
     => ( member_c @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_345_vimageD,axiom,
    ! [A2: set_d,F: set_d > set_d,A: set_set_d] :
      ( ( member_set_d @ A2 @ ( vimage_set_d_set_d @ F @ A ) )
     => ( member_set_d @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_346_measurable__sets__Collect,axiom,
    ! [F: c > c,M: sigma_measure_c,N: sigma_measure_c,P: c > $o] :
      ( ( member_c_c @ F @ ( sigma_measurable_c_c @ M @ N ) )
     => ( ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ N ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_c @ N ) )
       => ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( P @ ( F @ X2 ) ) ) )
          @ ( sigma_sets_c @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_347_measurable__sets__Collect,axiom,
    ! [F: b > c,M: sigma_measure_b,N: sigma_measure_c,P: c > $o] :
      ( ( member_b_c @ F @ ( sigma_measurable_b_c @ M @ N ) )
     => ( ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ N ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_c @ N ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( P @ ( F @ X2 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_348_measurable__sets__Collect,axiom,
    ! [F: d > c,M: sigma_measure_d,N: sigma_measure_c,P: c > $o] :
      ( ( member_d_c @ F @ ( sigma_measurable_d_c @ M @ N ) )
     => ( ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ N ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_c @ N ) )
       => ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( P @ ( F @ X2 ) ) ) )
          @ ( sigma_sets_d @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_349_measurable__sets__Collect,axiom,
    ! [F: c > b,M: sigma_measure_c,N: sigma_measure_b,P: b > $o] :
      ( ( member_c_b @ F @ ( sigma_measurable_c_b @ M @ N ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ N ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b @ N ) )
       => ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( P @ ( F @ X2 ) ) ) )
          @ ( sigma_sets_c @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_350_measurable__sets__Collect,axiom,
    ! [F: c > d,M: sigma_measure_c,N: sigma_measure_d,P: d > $o] :
      ( ( member_c_d @ F @ ( sigma_measurable_c_d @ M @ N ) )
     => ( ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ N ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d @ N ) )
       => ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( P @ ( F @ X2 ) ) ) )
          @ ( sigma_sets_c @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_351_measurable__sets__Collect,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d,P: d > $o] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
     => ( ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ N ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d @ N ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( P @ ( F @ X2 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_352_measurable__sets__Collect,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,P: b > $o] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ N ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b @ N ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( P @ ( F @ X2 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_353_measurable__sets__Collect,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b,P: b > $o] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ N ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b @ N ) )
       => ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( P @ ( F @ X2 ) ) ) )
          @ ( sigma_sets_d @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_354_measurable__sets__Collect,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d,P: d > $o] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
     => ( ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ N ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d @ N ) )
       => ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( P @ ( F @ X2 ) ) ) )
          @ ( sigma_sets_d @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_355_measurable__sets__Collect,axiom,
    ! [F: c > set_d,M: sigma_measure_c,N: sigma_measure_set_d,P: set_d > $o] :
      ( ( member_c_set_d @ F @ ( sigma_6374063253281460273_set_d @ M @ N ) )
     => ( ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ N ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_set_d @ N ) )
       => ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( P @ ( F @ X2 ) ) ) )
          @ ( sigma_sets_c @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_356_measurable__If,axiom,
    ! [F: b > d,M: sigma_measure_b,M2: sigma_measure_d,G: b > d,P: b > $o] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ M2 ) )
     => ( ( member_b_d @ G @ ( sigma_measurable_b_d @ M @ M2 ) )
       => ( ( member_set_b
            @ ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                  & ( P @ X2 ) ) )
            @ ( sigma_sets_b @ M ) )
         => ( member_b_d
            @ ^ [X2: b] : ( if_d @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
            @ ( sigma_measurable_b_d @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_357_measurable__If,axiom,
    ! [F: b > b,M: sigma_measure_b,M2: sigma_measure_b,G: b > b,P: b > $o] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ M2 ) )
     => ( ( member_b_b @ G @ ( sigma_measurable_b_b @ M @ M2 ) )
       => ( ( member_set_b
            @ ( collect_b
              @ ^ [X2: b] :
                  ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                  & ( P @ X2 ) ) )
            @ ( sigma_sets_b @ M ) )
         => ( member_b_b
            @ ^ [X2: b] : ( if_b @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
            @ ( sigma_measurable_b_b @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_358_measurable__If,axiom,
    ! [F: d > b,M: sigma_measure_d,M2: sigma_measure_b,G: d > b,P: d > $o] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ M2 ) )
     => ( ( member_d_b @ G @ ( sigma_measurable_d_b @ M @ M2 ) )
       => ( ( member_set_d
            @ ( collect_d
              @ ^ [X2: d] :
                  ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                  & ( P @ X2 ) ) )
            @ ( sigma_sets_d @ M ) )
         => ( member_d_b
            @ ^ [X2: d] : ( if_b @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
            @ ( sigma_measurable_d_b @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_359_measurable__If,axiom,
    ! [F: d > d,M: sigma_measure_d,M2: sigma_measure_d,G: d > d,P: d > $o] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ M2 ) )
     => ( ( member_d_d @ G @ ( sigma_measurable_d_d @ M @ M2 ) )
       => ( ( member_set_d
            @ ( collect_d
              @ ^ [X2: d] :
                  ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                  & ( P @ X2 ) ) )
            @ ( sigma_sets_d @ M ) )
         => ( member_d_d
            @ ^ [X2: d] : ( if_d @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
            @ ( sigma_measurable_d_d @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_360_Collect__subset,axiom,
    ! [A: set_set_d,P: set_d > $o] :
      ( ord_le3937772867789547105_set_d
      @ ( collect_set_d
        @ ^ [X2: set_d] :
            ( ( member_set_d @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_361_Collect__subset,axiom,
    ! [A: set_c,P: c > $o] :
      ( ord_less_eq_set_c
      @ ( collect_c
        @ ^ [X2: c] :
            ( ( member_c @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_362_Collect__subset,axiom,
    ! [A: set_b_d,P: ( b > d ) > $o] :
      ( ord_less_eq_set_b_d
      @ ( collect_b_d
        @ ^ [X2: b > d] :
            ( ( member_b_d @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_363_Collect__subset,axiom,
    ! [A: set_b_b,P: ( b > b ) > $o] :
      ( ord_less_eq_set_b_b
      @ ( collect_b_b
        @ ^ [X2: b > b] :
            ( ( member_b_b @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_364_Collect__subset,axiom,
    ! [A: set_d_b,P: ( d > b ) > $o] :
      ( ord_less_eq_set_d_b
      @ ( collect_d_b
        @ ^ [X2: d > b] :
            ( ( member_d_b @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_365_Collect__subset,axiom,
    ! [A: set_d_d,P: ( d > d ) > $o] :
      ( ord_less_eq_set_d_d
      @ ( collect_d_d
        @ ^ [X2: d > d] :
            ( ( member_d_d @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_366_Collect__subset,axiom,
    ! [A: set_set_b,P: set_b > $o] :
      ( ord_le3795704787696855135_set_b
      @ ( collect_set_b
        @ ^ [X2: set_b] :
            ( ( member_set_b @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_367_Collect__subset,axiom,
    ! [A: set_b,P: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X2: b] :
            ( ( member_b @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_368_Collect__conj__eq,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ( collect_b
        @ ^ [X2: b] :
            ( ( P @ X2 )
            & ( Q @ X2 ) ) )
      = ( inf_inf_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_369_Collect__conj__eq,axiom,
    ! [P: d > $o,Q: d > $o] :
      ( ( collect_d
        @ ^ [X2: d] :
            ( ( P @ X2 )
            & ( Q @ X2 ) ) )
      = ( inf_inf_set_d @ ( collect_d @ P ) @ ( collect_d @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_370_Collect__conj__eq,axiom,
    ! [P: set_b > $o,Q: set_b > $o] :
      ( ( collect_set_b
        @ ^ [X2: set_b] :
            ( ( P @ X2 )
            & ( Q @ X2 ) ) )
      = ( inf_inf_set_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_371_Int__Collect,axiom,
    ! [X: set_d,A: set_set_d,P: set_d > $o] :
      ( ( member_set_d @ X @ ( inf_inf_set_set_d @ A @ ( collect_set_d @ P ) ) )
      = ( ( member_set_d @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_372_Int__Collect,axiom,
    ! [X: c,A: set_c,P: c > $o] :
      ( ( member_c @ X @ ( inf_inf_set_c @ A @ ( collect_c @ P ) ) )
      = ( ( member_c @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_373_Int__Collect,axiom,
    ! [X: b > d,A: set_b_d,P: ( b > d ) > $o] :
      ( ( member_b_d @ X @ ( inf_inf_set_b_d @ A @ ( collect_b_d @ P ) ) )
      = ( ( member_b_d @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_374_Int__Collect,axiom,
    ! [X: b > b,A: set_b_b,P: ( b > b ) > $o] :
      ( ( member_b_b @ X @ ( inf_inf_set_b_b @ A @ ( collect_b_b @ P ) ) )
      = ( ( member_b_b @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_375_Int__Collect,axiom,
    ! [X: d > b,A: set_d_b,P: ( d > b ) > $o] :
      ( ( member_d_b @ X @ ( inf_inf_set_d_b @ A @ ( collect_d_b @ P ) ) )
      = ( ( member_d_b @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_376_Int__Collect,axiom,
    ! [X: d > d,A: set_d_d,P: ( d > d ) > $o] :
      ( ( member_d_d @ X @ ( inf_inf_set_d_d @ A @ ( collect_d_d @ P ) ) )
      = ( ( member_d_d @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_377_Int__Collect,axiom,
    ! [X: b,A: set_b,P: b > $o] :
      ( ( member_b @ X @ ( inf_inf_set_b @ A @ ( collect_b @ P ) ) )
      = ( ( member_b @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_378_Int__Collect,axiom,
    ! [X: d,A: set_d,P: d > $o] :
      ( ( member_d @ X @ ( inf_inf_set_d @ A @ ( collect_d @ P ) ) )
      = ( ( member_d @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_379_Int__Collect,axiom,
    ! [X: set_b,A: set_set_b,P: set_b > $o] :
      ( ( member_set_b @ X @ ( inf_inf_set_set_b @ A @ ( collect_set_b @ P ) ) )
      = ( ( member_set_b @ X @ A )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_380_Int__def,axiom,
    ( inf_inf_set_set_d
    = ( ^ [A3: set_set_d,B3: set_set_d] :
          ( collect_set_d
          @ ^ [X2: set_d] :
              ( ( member_set_d @ X2 @ A3 )
              & ( member_set_d @ X2 @ B3 ) ) ) ) ) ).

% Int_def
thf(fact_381_Int__def,axiom,
    ( inf_inf_set_c
    = ( ^ [A3: set_c,B3: set_c] :
          ( collect_c
          @ ^ [X2: c] :
              ( ( member_c @ X2 @ A3 )
              & ( member_c @ X2 @ B3 ) ) ) ) ) ).

% Int_def
thf(fact_382_Int__def,axiom,
    ( inf_inf_set_b_d
    = ( ^ [A3: set_b_d,B3: set_b_d] :
          ( collect_b_d
          @ ^ [X2: b > d] :
              ( ( member_b_d @ X2 @ A3 )
              & ( member_b_d @ X2 @ B3 ) ) ) ) ) ).

% Int_def
thf(fact_383_Int__def,axiom,
    ( inf_inf_set_b_b
    = ( ^ [A3: set_b_b,B3: set_b_b] :
          ( collect_b_b
          @ ^ [X2: b > b] :
              ( ( member_b_b @ X2 @ A3 )
              & ( member_b_b @ X2 @ B3 ) ) ) ) ) ).

% Int_def
thf(fact_384_Int__def,axiom,
    ( inf_inf_set_d_b
    = ( ^ [A3: set_d_b,B3: set_d_b] :
          ( collect_d_b
          @ ^ [X2: d > b] :
              ( ( member_d_b @ X2 @ A3 )
              & ( member_d_b @ X2 @ B3 ) ) ) ) ) ).

% Int_def
thf(fact_385_Int__def,axiom,
    ( inf_inf_set_d_d
    = ( ^ [A3: set_d_d,B3: set_d_d] :
          ( collect_d_d
          @ ^ [X2: d > d] :
              ( ( member_d_d @ X2 @ A3 )
              & ( member_d_d @ X2 @ B3 ) ) ) ) ) ).

% Int_def
thf(fact_386_Int__def,axiom,
    ( inf_inf_set_b
    = ( ^ [A3: set_b,B3: set_b] :
          ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ A3 )
              & ( member_b @ X2 @ B3 ) ) ) ) ) ).

% Int_def
thf(fact_387_Int__def,axiom,
    ( inf_inf_set_d
    = ( ^ [A3: set_d,B3: set_d] :
          ( collect_d
          @ ^ [X2: d] :
              ( ( member_d @ X2 @ A3 )
              & ( member_d @ X2 @ B3 ) ) ) ) ) ).

% Int_def
thf(fact_388_Int__def,axiom,
    ( inf_inf_set_set_b
    = ( ^ [A3: set_set_b,B3: set_set_b] :
          ( collect_set_b
          @ ^ [X2: set_b] :
              ( ( member_set_b @ X2 @ A3 )
              & ( member_set_b @ X2 @ B3 ) ) ) ) ) ).

% Int_def
thf(fact_389_vimage__def,axiom,
    ( vimage_b_d
    = ( ^ [F2: b > d,B3: set_d] :
          ( collect_b
          @ ^ [X2: b] : ( member_d @ ( F2 @ X2 ) @ B3 ) ) ) ) ).

% vimage_def
thf(fact_390_vimage__def,axiom,
    ( vimage_b_b
    = ( ^ [F2: b > b,B3: set_b] :
          ( collect_b
          @ ^ [X2: b] : ( member_b @ ( F2 @ X2 ) @ B3 ) ) ) ) ).

% vimage_def
thf(fact_391_vimage__def,axiom,
    ( vimage_d_b
    = ( ^ [F2: d > b,B3: set_b] :
          ( collect_d
          @ ^ [X2: d] : ( member_b @ ( F2 @ X2 ) @ B3 ) ) ) ) ).

% vimage_def
thf(fact_392_vimage__def,axiom,
    ( vimage_d_d
    = ( ^ [F2: d > d,B3: set_d] :
          ( collect_d
          @ ^ [X2: d] : ( member_d @ ( F2 @ X2 ) @ B3 ) ) ) ) ).

% vimage_def
thf(fact_393_vimage__def,axiom,
    ( vimage_set_b_c
    = ( ^ [F2: set_b > c,B3: set_c] :
          ( collect_set_b
          @ ^ [X2: set_b] : ( member_c @ ( F2 @ X2 ) @ B3 ) ) ) ) ).

% vimage_def
thf(fact_394_vimage__def,axiom,
    ( vimage_set_b_set_d
    = ( ^ [F2: set_b > set_d,B3: set_set_d] :
          ( collect_set_b
          @ ^ [X2: set_b] : ( member_set_d @ ( F2 @ X2 ) @ B3 ) ) ) ) ).

% vimage_def
thf(fact_395_vimage__def,axiom,
    ( vimage_set_b_set_b
    = ( ^ [F2: set_b > set_b,B3: set_set_b] :
          ( collect_set_b
          @ ^ [X2: set_b] : ( member_set_b @ ( F2 @ X2 ) @ B3 ) ) ) ) ).

% vimage_def
thf(fact_396_vimage__def,axiom,
    ( vimage_set_b_b_d
    = ( ^ [F2: set_b > b > d,B3: set_b_d] :
          ( collect_set_b
          @ ^ [X2: set_b] : ( member_b_d @ ( F2 @ X2 ) @ B3 ) ) ) ) ).

% vimage_def
thf(fact_397_vimage__def,axiom,
    ( vimage_set_b_b_b
    = ( ^ [F2: set_b > b > b,B3: set_b_b] :
          ( collect_set_b
          @ ^ [X2: set_b] : ( member_b_b @ ( F2 @ X2 ) @ B3 ) ) ) ) ).

% vimage_def
thf(fact_398_vimage__def,axiom,
    ( vimage_set_b_d_b
    = ( ^ [F2: set_b > d > b,B3: set_d_b] :
          ( collect_set_b
          @ ^ [X2: set_b] : ( member_d_b @ ( F2 @ X2 ) @ B3 ) ) ) ) ).

% vimage_def
thf(fact_399_measurable__If__set,axiom,
    ! [F: b > d,M: sigma_measure_b,M2: sigma_measure_d,G: b > d,A: set_b] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ M2 ) )
     => ( ( member_b_d @ G @ ( sigma_measurable_b_d @ M @ M2 ) )
       => ( ( member_set_b @ ( inf_inf_set_b @ A @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
         => ( member_b_d
            @ ^ [X2: b] : ( if_d @ ( member_b @ X2 @ A ) @ ( F @ X2 ) @ ( G @ X2 ) )
            @ ( sigma_measurable_b_d @ M @ M2 ) ) ) ) ) ).

% measurable_If_set
thf(fact_400_measurable__If__set,axiom,
    ! [F: b > b,M: sigma_measure_b,M2: sigma_measure_b,G: b > b,A: set_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ M2 ) )
     => ( ( member_b_b @ G @ ( sigma_measurable_b_b @ M @ M2 ) )
       => ( ( member_set_b @ ( inf_inf_set_b @ A @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
         => ( member_b_b
            @ ^ [X2: b] : ( if_b @ ( member_b @ X2 @ A ) @ ( F @ X2 ) @ ( G @ X2 ) )
            @ ( sigma_measurable_b_b @ M @ M2 ) ) ) ) ) ).

% measurable_If_set
thf(fact_401_measurable__If__set,axiom,
    ! [F: d > b,M: sigma_measure_d,M2: sigma_measure_b,G: d > b,A: set_d] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ M2 ) )
     => ( ( member_d_b @ G @ ( sigma_measurable_d_b @ M @ M2 ) )
       => ( ( member_set_d @ ( inf_inf_set_d @ A @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) )
         => ( member_d_b
            @ ^ [X2: d] : ( if_b @ ( member_d @ X2 @ A ) @ ( F @ X2 ) @ ( G @ X2 ) )
            @ ( sigma_measurable_d_b @ M @ M2 ) ) ) ) ) ).

% measurable_If_set
thf(fact_402_measurable__If__set,axiom,
    ! [F: d > d,M: sigma_measure_d,M2: sigma_measure_d,G: d > d,A: set_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ M2 ) )
     => ( ( member_d_d @ G @ ( sigma_measurable_d_d @ M @ M2 ) )
       => ( ( member_set_d @ ( inf_inf_set_d @ A @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) )
         => ( member_d_d
            @ ^ [X2: d] : ( if_d @ ( member_d @ X2 @ A ) @ ( F @ X2 ) @ ( G @ X2 ) )
            @ ( sigma_measurable_d_d @ M @ M2 ) ) ) ) ) ).

% measurable_If_set
thf(fact_403_inf_OcoboundedI2,axiom,
    ! [B2: set_d,C2: set_d,A2: set_d] :
      ( ( ord_less_eq_set_d @ B2 @ C2 )
     => ( ord_less_eq_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_404_inf_OcoboundedI2,axiom,
    ! [B2: set_set_b,C2: set_set_b,A2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
     => ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_405_inf_OcoboundedI2,axiom,
    ! [B2: set_b,C2: set_b,A2: set_b] :
      ( ( ord_less_eq_set_b @ B2 @ C2 )
     => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_406_inf_OcoboundedI1,axiom,
    ! [A2: set_d,C2: set_d,B2: set_d] :
      ( ( ord_less_eq_set_d @ A2 @ C2 )
     => ( ord_less_eq_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_407_inf_OcoboundedI1,axiom,
    ! [A2: set_set_b,C2: set_set_b,B2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ C2 )
     => ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_408_inf_OcoboundedI1,axiom,
    ! [A2: set_b,C2: set_b,B2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ C2 )
     => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_409_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_d
    = ( ^ [B4: set_d,A4: set_d] :
          ( ( inf_inf_set_d @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_410_inf_Oabsorb__iff2,axiom,
    ( ord_le3795704787696855135_set_b
    = ( ^ [B4: set_set_b,A4: set_set_b] :
          ( ( inf_inf_set_set_b @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_411_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_b
    = ( ^ [B4: set_b,A4: set_b] :
          ( ( inf_inf_set_b @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_412_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_d
    = ( ^ [A4: set_d,B4: set_d] :
          ( ( inf_inf_set_d @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_413_inf_Oabsorb__iff1,axiom,
    ( ord_le3795704787696855135_set_b
    = ( ^ [A4: set_set_b,B4: set_set_b] :
          ( ( inf_inf_set_set_b @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_414_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A4: set_b,B4: set_b] :
          ( ( inf_inf_set_b @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_415_inf_Ocobounded2,axiom,
    ! [A2: set_d,B2: set_d] : ( ord_less_eq_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_416_inf_Ocobounded2,axiom,
    ! [A2: set_set_b,B2: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_417_inf_Ocobounded2,axiom,
    ! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 ) ).

% inf.cobounded2
thf(fact_418_inf_Ocobounded1,axiom,
    ! [A2: set_d,B2: set_d] : ( ord_less_eq_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_419_inf_Ocobounded1,axiom,
    ! [A2: set_set_b,B2: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_420_inf_Ocobounded1,axiom,
    ! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ A2 ) ).

% inf.cobounded1
thf(fact_421_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_d
    = ( ^ [A4: set_d,B4: set_d] :
          ( A4
          = ( inf_inf_set_d @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_422_inf_Oorder__iff,axiom,
    ( ord_le3795704787696855135_set_b
    = ( ^ [A4: set_set_b,B4: set_set_b] :
          ( A4
          = ( inf_inf_set_set_b @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_423_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A4: set_b,B4: set_b] :
          ( A4
          = ( inf_inf_set_b @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_424_inf__greatest,axiom,
    ! [X: set_d,Y: set_d,Z: set_d] :
      ( ( ord_less_eq_set_d @ X @ Y )
     => ( ( ord_less_eq_set_d @ X @ Z )
       => ( ord_less_eq_set_d @ X @ ( inf_inf_set_d @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_425_inf__greatest,axiom,
    ! [X: set_set_b,Y: set_set_b,Z: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ X @ Y )
     => ( ( ord_le3795704787696855135_set_b @ X @ Z )
       => ( ord_le3795704787696855135_set_b @ X @ ( inf_inf_set_set_b @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_426_inf__greatest,axiom,
    ! [X: set_b,Y: set_b,Z: set_b] :
      ( ( ord_less_eq_set_b @ X @ Y )
     => ( ( ord_less_eq_set_b @ X @ Z )
       => ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_427_inf_OboundedI,axiom,
    ! [A2: set_d,B2: set_d,C2: set_d] :
      ( ( ord_less_eq_set_d @ A2 @ B2 )
     => ( ( ord_less_eq_set_d @ A2 @ C2 )
       => ( ord_less_eq_set_d @ A2 @ ( inf_inf_set_d @ B2 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_428_inf_OboundedI,axiom,
    ! [A2: set_set_b,B2: set_set_b,C2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( ( ord_le3795704787696855135_set_b @ A2 @ C2 )
       => ( ord_le3795704787696855135_set_b @ A2 @ ( inf_inf_set_set_b @ B2 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_429_inf_OboundedI,axiom,
    ! [A2: set_b,B2: set_b,C2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( ( ord_less_eq_set_b @ A2 @ C2 )
       => ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_430_inf_OboundedE,axiom,
    ! [A2: set_d,B2: set_d,C2: set_d] :
      ( ( ord_less_eq_set_d @ A2 @ ( inf_inf_set_d @ B2 @ C2 ) )
     => ~ ( ( ord_less_eq_set_d @ A2 @ B2 )
         => ~ ( ord_less_eq_set_d @ A2 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_431_inf_OboundedE,axiom,
    ! [A2: set_set_b,B2: set_set_b,C2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ ( inf_inf_set_set_b @ B2 @ C2 ) )
     => ~ ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
         => ~ ( ord_le3795704787696855135_set_b @ A2 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_432_inf_OboundedE,axiom,
    ! [A2: set_b,B2: set_b,C2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ ( inf_inf_set_b @ B2 @ C2 ) )
     => ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
         => ~ ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).

% inf.boundedE
thf(fact_433_inf__absorb2,axiom,
    ! [Y: set_d,X: set_d] :
      ( ( ord_less_eq_set_d @ Y @ X )
     => ( ( inf_inf_set_d @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_434_inf__absorb2,axiom,
    ! [Y: set_set_b,X: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ Y @ X )
     => ( ( inf_inf_set_set_b @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_435_inf__absorb2,axiom,
    ! [Y: set_b,X: set_b] :
      ( ( ord_less_eq_set_b @ Y @ X )
     => ( ( inf_inf_set_b @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_436_inf__absorb1,axiom,
    ! [X: set_d,Y: set_d] :
      ( ( ord_less_eq_set_d @ X @ Y )
     => ( ( inf_inf_set_d @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_437_inf__absorb1,axiom,
    ! [X: set_set_b,Y: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ X @ Y )
     => ( ( inf_inf_set_set_b @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_438_inf__absorb1,axiom,
    ! [X: set_b,Y: set_b] :
      ( ( ord_less_eq_set_b @ X @ Y )
     => ( ( inf_inf_set_b @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_439_inf_Oabsorb2,axiom,
    ! [B2: set_d,A2: set_d] :
      ( ( ord_less_eq_set_d @ B2 @ A2 )
     => ( ( inf_inf_set_d @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_440_inf_Oabsorb2,axiom,
    ! [B2: set_set_b,A2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ B2 @ A2 )
     => ( ( inf_inf_set_set_b @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_441_inf_Oabsorb2,axiom,
    ! [B2: set_b,A2: set_b] :
      ( ( ord_less_eq_set_b @ B2 @ A2 )
     => ( ( inf_inf_set_b @ A2 @ B2 )
        = B2 ) ) ).

% inf.absorb2
thf(fact_442_inf_Oabsorb1,axiom,
    ! [A2: set_d,B2: set_d] :
      ( ( ord_less_eq_set_d @ A2 @ B2 )
     => ( ( inf_inf_set_d @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_443_inf_Oabsorb1,axiom,
    ! [A2: set_set_b,B2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( ( inf_inf_set_set_b @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_444_inf_Oabsorb1,axiom,
    ! [A2: set_b,B2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( ( inf_inf_set_b @ A2 @ B2 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_445_le__iff__inf,axiom,
    ( ord_less_eq_set_d
    = ( ^ [X2: set_d,Y3: set_d] :
          ( ( inf_inf_set_d @ X2 @ Y3 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_446_le__iff__inf,axiom,
    ( ord_le3795704787696855135_set_b
    = ( ^ [X2: set_set_b,Y3: set_set_b] :
          ( ( inf_inf_set_set_b @ X2 @ Y3 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_447_le__iff__inf,axiom,
    ( ord_less_eq_set_b
    = ( ^ [X2: set_b,Y3: set_b] :
          ( ( inf_inf_set_b @ X2 @ Y3 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_448_inf__unique,axiom,
    ! [F: set_d > set_d > set_d,X: set_d,Y: set_d] :
      ( ! [X3: set_d,Y5: set_d] : ( ord_less_eq_set_d @ ( F @ X3 @ Y5 ) @ X3 )
     => ( ! [X3: set_d,Y5: set_d] : ( ord_less_eq_set_d @ ( F @ X3 @ Y5 ) @ Y5 )
       => ( ! [X3: set_d,Y5: set_d,Z3: set_d] :
              ( ( ord_less_eq_set_d @ X3 @ Y5 )
             => ( ( ord_less_eq_set_d @ X3 @ Z3 )
               => ( ord_less_eq_set_d @ X3 @ ( F @ Y5 @ Z3 ) ) ) )
         => ( ( inf_inf_set_d @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_449_inf__unique,axiom,
    ! [F: set_set_b > set_set_b > set_set_b,X: set_set_b,Y: set_set_b] :
      ( ! [X3: set_set_b,Y5: set_set_b] : ( ord_le3795704787696855135_set_b @ ( F @ X3 @ Y5 ) @ X3 )
     => ( ! [X3: set_set_b,Y5: set_set_b] : ( ord_le3795704787696855135_set_b @ ( F @ X3 @ Y5 ) @ Y5 )
       => ( ! [X3: set_set_b,Y5: set_set_b,Z3: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ X3 @ Y5 )
             => ( ( ord_le3795704787696855135_set_b @ X3 @ Z3 )
               => ( ord_le3795704787696855135_set_b @ X3 @ ( F @ Y5 @ Z3 ) ) ) )
         => ( ( inf_inf_set_set_b @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_450_inf__unique,axiom,
    ! [F: set_b > set_b > set_b,X: set_b,Y: set_b] :
      ( ! [X3: set_b,Y5: set_b] : ( ord_less_eq_set_b @ ( F @ X3 @ Y5 ) @ X3 )
     => ( ! [X3: set_b,Y5: set_b] : ( ord_less_eq_set_b @ ( F @ X3 @ Y5 ) @ Y5 )
       => ( ! [X3: set_b,Y5: set_b,Z3: set_b] :
              ( ( ord_less_eq_set_b @ X3 @ Y5 )
             => ( ( ord_less_eq_set_b @ X3 @ Z3 )
               => ( ord_less_eq_set_b @ X3 @ ( F @ Y5 @ Z3 ) ) ) )
         => ( ( inf_inf_set_b @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_451_inf_OorderI,axiom,
    ! [A2: set_d,B2: set_d] :
      ( ( A2
        = ( inf_inf_set_d @ A2 @ B2 ) )
     => ( ord_less_eq_set_d @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_452_inf_OorderI,axiom,
    ! [A2: set_set_b,B2: set_set_b] :
      ( ( A2
        = ( inf_inf_set_set_b @ A2 @ B2 ) )
     => ( ord_le3795704787696855135_set_b @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_453_inf_OorderI,axiom,
    ! [A2: set_b,B2: set_b] :
      ( ( A2
        = ( inf_inf_set_b @ A2 @ B2 ) )
     => ( ord_less_eq_set_b @ A2 @ B2 ) ) ).

% inf.orderI
thf(fact_454_inf_OorderE,axiom,
    ! [A2: set_d,B2: set_d] :
      ( ( ord_less_eq_set_d @ A2 @ B2 )
     => ( A2
        = ( inf_inf_set_d @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_455_inf_OorderE,axiom,
    ! [A2: set_set_b,B2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( A2
        = ( inf_inf_set_set_b @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_456_inf_OorderE,axiom,
    ! [A2: set_b,B2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( A2
        = ( inf_inf_set_b @ A2 @ B2 ) ) ) ).

% inf.orderE
thf(fact_457_le__infI2,axiom,
    ! [B2: set_d,X: set_d,A2: set_d] :
      ( ( ord_less_eq_set_d @ B2 @ X )
     => ( ord_less_eq_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_458_le__infI2,axiom,
    ! [B2: set_set_b,X: set_set_b,A2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ B2 @ X )
     => ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_459_le__infI2,axiom,
    ! [B2: set_b,X: set_b,A2: set_b] :
      ( ( ord_less_eq_set_b @ B2 @ X )
     => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X ) ) ).

% le_infI2
thf(fact_460_le__infI1,axiom,
    ! [A2: set_d,X: set_d,B2: set_d] :
      ( ( ord_less_eq_set_d @ A2 @ X )
     => ( ord_less_eq_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_461_le__infI1,axiom,
    ! [A2: set_set_b,X: set_set_b,B2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ X )
     => ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_462_le__infI1,axiom,
    ! [A2: set_b,X: set_b,B2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ X )
     => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ X ) ) ).

% le_infI1
thf(fact_463_inf__mono,axiom,
    ! [A2: set_d,C2: set_d,B2: set_d,D: set_d] :
      ( ( ord_less_eq_set_d @ A2 @ C2 )
     => ( ( ord_less_eq_set_d @ B2 @ D )
       => ( ord_less_eq_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ ( inf_inf_set_d @ C2 @ D ) ) ) ) ).

% inf_mono
thf(fact_464_inf__mono,axiom,
    ! [A2: set_set_b,C2: set_set_b,B2: set_set_b,D: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ C2 )
     => ( ( ord_le3795704787696855135_set_b @ B2 @ D )
       => ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ ( inf_inf_set_set_b @ C2 @ D ) ) ) ) ).

% inf_mono
thf(fact_465_inf__mono,axiom,
    ! [A2: set_b,C2: set_b,B2: set_b,D: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ C2 )
     => ( ( ord_less_eq_set_b @ B2 @ D )
       => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( inf_inf_set_b @ C2 @ D ) ) ) ) ).

% inf_mono
thf(fact_466_le__infI,axiom,
    ! [X: set_d,A2: set_d,B2: set_d] :
      ( ( ord_less_eq_set_d @ X @ A2 )
     => ( ( ord_less_eq_set_d @ X @ B2 )
       => ( ord_less_eq_set_d @ X @ ( inf_inf_set_d @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_467_le__infI,axiom,
    ! [X: set_set_b,A2: set_set_b,B2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ X @ A2 )
     => ( ( ord_le3795704787696855135_set_b @ X @ B2 )
       => ( ord_le3795704787696855135_set_b @ X @ ( inf_inf_set_set_b @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_468_le__infI,axiom,
    ! [X: set_b,A2: set_b,B2: set_b] :
      ( ( ord_less_eq_set_b @ X @ A2 )
     => ( ( ord_less_eq_set_b @ X @ B2 )
       => ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).

% le_infI
thf(fact_469_le__infE,axiom,
    ! [X: set_d,A2: set_d,B2: set_d] :
      ( ( ord_less_eq_set_d @ X @ ( inf_inf_set_d @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_set_d @ X @ A2 )
         => ~ ( ord_less_eq_set_d @ X @ B2 ) ) ) ).

% le_infE
thf(fact_470_le__infE,axiom,
    ! [X: set_set_b,A2: set_set_b,B2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ X @ ( inf_inf_set_set_b @ A2 @ B2 ) )
     => ~ ( ( ord_le3795704787696855135_set_b @ X @ A2 )
         => ~ ( ord_le3795704787696855135_set_b @ X @ B2 ) ) ) ).

% le_infE
thf(fact_471_le__infE,axiom,
    ! [X: set_b,A2: set_b,B2: set_b] :
      ( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A2 @ B2 ) )
     => ~ ( ( ord_less_eq_set_b @ X @ A2 )
         => ~ ( ord_less_eq_set_b @ X @ B2 ) ) ) ).

% le_infE
thf(fact_472_inf__le2,axiom,
    ! [X: set_d,Y: set_d] : ( ord_less_eq_set_d @ ( inf_inf_set_d @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_473_inf__le2,axiom,
    ! [X: set_set_b,Y: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_474_inf__le2,axiom,
    ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_475_inf__le1,axiom,
    ! [X: set_d,Y: set_d] : ( ord_less_eq_set_d @ ( inf_inf_set_d @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_476_inf__le1,axiom,
    ! [X: set_set_b,Y: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_477_inf__le1,axiom,
    ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_478_inf__sup__ord_I1_J,axiom,
    ! [X: set_d,Y: set_d] : ( ord_less_eq_set_d @ ( inf_inf_set_d @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_479_inf__sup__ord_I1_J,axiom,
    ! [X: set_set_b,Y: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_480_inf__sup__ord_I1_J,axiom,
    ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_481_inf__sup__ord_I2_J,axiom,
    ! [X: set_d,Y: set_d] : ( ord_less_eq_set_d @ ( inf_inf_set_d @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_482_inf__sup__ord_I2_J,axiom,
    ! [X: set_set_b,Y: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_483_inf__sup__ord_I2_J,axiom,
    ! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_484_Int__Collect__mono,axiom,
    ! [A: set_set_d,B: set_set_d,P: set_d > $o,Q: set_d > $o] :
      ( ( ord_le3937772867789547105_set_d @ A @ B )
     => ( ! [X3: set_d] :
            ( ( member_set_d @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le3937772867789547105_set_d @ ( inf_inf_set_set_d @ A @ ( collect_set_d @ P ) ) @ ( inf_inf_set_set_d @ B @ ( collect_set_d @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_485_Int__Collect__mono,axiom,
    ! [A: set_c,B: set_c,P: c > $o,Q: c > $o] :
      ( ( ord_less_eq_set_c @ A @ B )
     => ( ! [X3: c] :
            ( ( member_c @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_c @ ( inf_inf_set_c @ A @ ( collect_c @ P ) ) @ ( inf_inf_set_c @ B @ ( collect_c @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_486_Int__Collect__mono,axiom,
    ! [A: set_b_d,B: set_b_d,P: ( b > d ) > $o,Q: ( b > d ) > $o] :
      ( ( ord_less_eq_set_b_d @ A @ B )
     => ( ! [X3: b > d] :
            ( ( member_b_d @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_b_d @ ( inf_inf_set_b_d @ A @ ( collect_b_d @ P ) ) @ ( inf_inf_set_b_d @ B @ ( collect_b_d @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_487_Int__Collect__mono,axiom,
    ! [A: set_b_b,B: set_b_b,P: ( b > b ) > $o,Q: ( b > b ) > $o] :
      ( ( ord_less_eq_set_b_b @ A @ B )
     => ( ! [X3: b > b] :
            ( ( member_b_b @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_b_b @ ( inf_inf_set_b_b @ A @ ( collect_b_b @ P ) ) @ ( inf_inf_set_b_b @ B @ ( collect_b_b @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_488_Int__Collect__mono,axiom,
    ! [A: set_d_b,B: set_d_b,P: ( d > b ) > $o,Q: ( d > b ) > $o] :
      ( ( ord_less_eq_set_d_b @ A @ B )
     => ( ! [X3: d > b] :
            ( ( member_d_b @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_d_b @ ( inf_inf_set_d_b @ A @ ( collect_d_b @ P ) ) @ ( inf_inf_set_d_b @ B @ ( collect_d_b @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_489_Int__Collect__mono,axiom,
    ! [A: set_d_d,B: set_d_d,P: ( d > d ) > $o,Q: ( d > d ) > $o] :
      ( ( ord_less_eq_set_d_d @ A @ B )
     => ( ! [X3: d > d] :
            ( ( member_d_d @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_d_d @ ( inf_inf_set_d_d @ A @ ( collect_d_d @ P ) ) @ ( inf_inf_set_d_d @ B @ ( collect_d_d @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_490_Int__Collect__mono,axiom,
    ! [A: set_d,B: set_d,P: d > $o,Q: d > $o] :
      ( ( ord_less_eq_set_d @ A @ B )
     => ( ! [X3: d] :
            ( ( member_d @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_d @ ( inf_inf_set_d @ A @ ( collect_d @ P ) ) @ ( inf_inf_set_d @ B @ ( collect_d @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_491_Int__Collect__mono,axiom,
    ! [A: set_set_b,B: set_set_b,P: set_b > $o,Q: set_b > $o] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ! [X3: set_b] :
            ( ( member_set_b @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ ( collect_set_b @ P ) ) @ ( inf_inf_set_set_b @ B @ ( collect_set_b @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_492_Int__Collect__mono,axiom,
    ! [A: set_b,B: set_b,P: b > $o,Q: b > $o] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ! [X3: b] :
            ( ( member_b @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B @ ( collect_b @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_493_Int__greatest,axiom,
    ! [C: set_d,A: set_d,B: set_d] :
      ( ( ord_less_eq_set_d @ C @ A )
     => ( ( ord_less_eq_set_d @ C @ B )
       => ( ord_less_eq_set_d @ C @ ( inf_inf_set_d @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_494_Int__greatest,axiom,
    ! [C: set_set_b,A: set_set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ C @ A )
     => ( ( ord_le3795704787696855135_set_b @ C @ B )
       => ( ord_le3795704787696855135_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_495_Int__greatest,axiom,
    ! [C: set_b,A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ C @ A )
     => ( ( ord_less_eq_set_b @ C @ B )
       => ( ord_less_eq_set_b @ C @ ( inf_inf_set_b @ A @ B ) ) ) ) ).

% Int_greatest
thf(fact_496_Int__absorb2,axiom,
    ! [A: set_d,B: set_d] :
      ( ( ord_less_eq_set_d @ A @ B )
     => ( ( inf_inf_set_d @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_497_Int__absorb2,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ( inf_inf_set_set_b @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_498_Int__absorb2,axiom,
    ! [A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( inf_inf_set_b @ A @ B )
        = A ) ) ).

% Int_absorb2
thf(fact_499_Int__absorb1,axiom,
    ! [B: set_d,A: set_d] :
      ( ( ord_less_eq_set_d @ B @ A )
     => ( ( inf_inf_set_d @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_500_Int__absorb1,axiom,
    ! [B: set_set_b,A: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ B @ A )
     => ( ( inf_inf_set_set_b @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_501_Int__absorb1,axiom,
    ! [B: set_b,A: set_b] :
      ( ( ord_less_eq_set_b @ B @ A )
     => ( ( inf_inf_set_b @ A @ B )
        = B ) ) ).

% Int_absorb1
thf(fact_502_Int__lower2,axiom,
    ! [A: set_d,B: set_d] : ( ord_less_eq_set_d @ ( inf_inf_set_d @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_503_Int__lower2,axiom,
    ! [A: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_504_Int__lower2,axiom,
    ! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ B ) ).

% Int_lower2
thf(fact_505_Int__lower1,axiom,
    ! [A: set_d,B: set_d] : ( ord_less_eq_set_d @ ( inf_inf_set_d @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_506_Int__lower1,axiom,
    ! [A: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_507_Int__lower1,axiom,
    ! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ A ) ).

% Int_lower1
thf(fact_508_Int__mono,axiom,
    ! [A: set_d,C: set_d,B: set_d,D2: set_d] :
      ( ( ord_less_eq_set_d @ A @ C )
     => ( ( ord_less_eq_set_d @ B @ D2 )
       => ( ord_less_eq_set_d @ ( inf_inf_set_d @ A @ B ) @ ( inf_inf_set_d @ C @ D2 ) ) ) ) ).

% Int_mono
thf(fact_509_Int__mono,axiom,
    ! [A: set_set_b,C: set_set_b,B: set_set_b,D2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ C )
     => ( ( ord_le3795704787696855135_set_b @ B @ D2 )
       => ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B ) @ ( inf_inf_set_set_b @ C @ D2 ) ) ) ) ).

% Int_mono
thf(fact_510_Int__mono,axiom,
    ! [A: set_b,C: set_b,B: set_b,D2: set_b] :
      ( ( ord_less_eq_set_b @ A @ C )
     => ( ( ord_less_eq_set_b @ B @ D2 )
       => ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ ( inf_inf_set_b @ C @ D2 ) ) ) ) ).

% Int_mono
thf(fact_511_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_b,M2: sigma_measure_b] :
      ( ( ( sigma_sets_b @ M )
        = ( sigma_sets_b @ M2 ) )
     => ( ( sigma_space_b @ M )
        = ( sigma_space_b @ M2 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_512_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_d,M2: sigma_measure_d] :
      ( ( ( sigma_sets_d @ M )
        = ( sigma_sets_d @ M2 ) )
     => ( ( sigma_space_d @ M )
        = ( sigma_space_d @ M2 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_513_subset__vimage__iff,axiom,
    ! [A: set_d,F: d > b,B: set_b] :
      ( ( ord_less_eq_set_d @ A @ ( vimage_d_b @ F @ B ) )
      = ( ! [X2: d] :
            ( ( member_d @ X2 @ A )
           => ( member_b @ ( F @ X2 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_514_subset__vimage__iff,axiom,
    ! [A: set_d,F: d > d,B: set_d] :
      ( ( ord_less_eq_set_d @ A @ ( vimage_d_d @ F @ B ) )
      = ( ! [X2: d] :
            ( ( member_d @ X2 @ A )
           => ( member_d @ ( F @ X2 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_515_subset__vimage__iff,axiom,
    ! [A: set_b,F: b > c,B: set_c] :
      ( ( ord_less_eq_set_b @ A @ ( vimage_b_c @ F @ B ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ( member_c @ ( F @ X2 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_516_subset__vimage__iff,axiom,
    ! [A: set_b,F: b > d,B: set_d] :
      ( ( ord_less_eq_set_b @ A @ ( vimage_b_d @ F @ B ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ( member_d @ ( F @ X2 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_517_subset__vimage__iff,axiom,
    ! [A: set_b,F: b > b,B: set_b] :
      ( ( ord_less_eq_set_b @ A @ ( vimage_b_b @ F @ B ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ( member_b @ ( F @ X2 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_518_subset__vimage__iff,axiom,
    ! [A: set_set_b,F: set_b > c,B: set_c] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( vimage_set_b_c @ F @ B ) )
      = ( ! [X2: set_b] :
            ( ( member_set_b @ X2 @ A )
           => ( member_c @ ( F @ X2 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_519_subset__vimage__iff,axiom,
    ! [A: set_b,F: b > set_d,B: set_set_d] :
      ( ( ord_less_eq_set_b @ A @ ( vimage_b_set_d @ F @ B ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ( member_set_d @ ( F @ X2 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_520_subset__vimage__iff,axiom,
    ! [A: set_b,F: b > set_b,B: set_set_b] :
      ( ( ord_less_eq_set_b @ A @ ( vimage_b_set_b @ F @ B ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ( member_set_b @ ( F @ X2 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_521_subset__vimage__iff,axiom,
    ! [A: set_set_b,F: set_b > set_d,B: set_set_d] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( vimage_set_b_set_d @ F @ B ) )
      = ( ! [X2: set_b] :
            ( ( member_set_b @ X2 @ A )
           => ( member_set_d @ ( F @ X2 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_522_subset__vimage__iff,axiom,
    ! [A: set_set_b,F: set_b > set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( vimage_set_b_set_b @ F @ B ) )
      = ( ! [X2: set_b] :
            ( ( member_set_b @ X2 @ A )
           => ( member_set_b @ ( F @ X2 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_523_vimage__mono,axiom,
    ! [A: set_d,B: set_d,F: d > d] :
      ( ( ord_less_eq_set_d @ A @ B )
     => ( ord_less_eq_set_d @ ( vimage_d_d @ F @ A ) @ ( vimage_d_d @ F @ B ) ) ) ).

% vimage_mono
thf(fact_524_vimage__mono,axiom,
    ! [A: set_d,B: set_d,F: b > d] :
      ( ( ord_less_eq_set_d @ A @ B )
     => ( ord_less_eq_set_b @ ( vimage_b_d @ F @ A ) @ ( vimage_b_d @ F @ B ) ) ) ).

% vimage_mono
thf(fact_525_vimage__mono,axiom,
    ! [A: set_set_b,B: set_set_b,F: set_b > set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ord_le3795704787696855135_set_b @ ( vimage_set_b_set_b @ F @ A ) @ ( vimage_set_b_set_b @ F @ B ) ) ) ).

% vimage_mono
thf(fact_526_vimage__mono,axiom,
    ! [A: set_set_b,B: set_set_b,F: b > set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ord_less_eq_set_b @ ( vimage_b_set_b @ F @ A ) @ ( vimage_b_set_b @ F @ B ) ) ) ).

% vimage_mono
thf(fact_527_vimage__mono,axiom,
    ! [A: set_b,B: set_b,F: d > b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ord_less_eq_set_d @ ( vimage_d_b @ F @ A ) @ ( vimage_d_b @ F @ B ) ) ) ).

% vimage_mono
thf(fact_528_vimage__mono,axiom,
    ! [A: set_b,B: set_b,F: set_b > b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ord_le3795704787696855135_set_b @ ( vimage_set_b_b @ F @ A ) @ ( vimage_set_b_b @ F @ B ) ) ) ).

% vimage_mono
thf(fact_529_vimage__mono,axiom,
    ! [A: set_b,B: set_b,F: b > b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ord_less_eq_set_b @ ( vimage_b_b @ F @ A ) @ ( vimage_b_b @ F @ B ) ) ) ).

% vimage_mono
thf(fact_530_vimage__inter__cong,axiom,
    ! [S: set_b,F: b > d,G: b > d,Y: set_d] :
      ( ! [W: b] :
          ( ( member_b @ W @ S )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( inf_inf_set_b @ ( vimage_b_d @ F @ Y ) @ S )
        = ( inf_inf_set_b @ ( vimage_b_d @ G @ Y ) @ S ) ) ) ).

% vimage_inter_cong
thf(fact_531_vimage__inter__cong,axiom,
    ! [S: set_b,F: b > b,G: b > b,Y: set_b] :
      ( ! [W: b] :
          ( ( member_b @ W @ S )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( inf_inf_set_b @ ( vimage_b_b @ F @ Y ) @ S )
        = ( inf_inf_set_b @ ( vimage_b_b @ G @ Y ) @ S ) ) ) ).

% vimage_inter_cong
thf(fact_532_vimage__inter__cong,axiom,
    ! [S: set_d,F: d > b,G: d > b,Y: set_b] :
      ( ! [W: d] :
          ( ( member_d @ W @ S )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( inf_inf_set_d @ ( vimage_d_b @ F @ Y ) @ S )
        = ( inf_inf_set_d @ ( vimage_d_b @ G @ Y ) @ S ) ) ) ).

% vimage_inter_cong
thf(fact_533_vimage__inter__cong,axiom,
    ! [S: set_d,F: d > d,G: d > d,Y: set_d] :
      ( ! [W: d] :
          ( ( member_d @ W @ S )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( inf_inf_set_d @ ( vimage_d_d @ F @ Y ) @ S )
        = ( inf_inf_set_d @ ( vimage_d_d @ G @ Y ) @ S ) ) ) ).

% vimage_inter_cong
thf(fact_534_measurableI,axiom,
    ! [M: sigma_measure_c,F: c > c,N: sigma_measure_c] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ ( sigma_space_c @ M ) )
         => ( member_c @ ( F @ X3 ) @ ( sigma_space_c @ N ) ) )
     => ( ! [A5: set_c] :
            ( ( member_set_c @ A5 @ ( sigma_sets_c @ N ) )
           => ( member_set_c @ ( inf_inf_set_c @ ( vimage_c_c @ F @ A5 ) @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) ) )
       => ( member_c_c @ F @ ( sigma_measurable_c_c @ M @ N ) ) ) ) ).

% measurableI
thf(fact_535_measurableI,axiom,
    ! [M: sigma_measure_c,F: c > b,N: sigma_measure_b] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ ( sigma_space_c @ M ) )
         => ( member_b @ ( F @ X3 ) @ ( sigma_space_b @ N ) ) )
     => ( ! [A5: set_b] :
            ( ( member_set_b @ A5 @ ( sigma_sets_b @ N ) )
           => ( member_set_c @ ( inf_inf_set_c @ ( vimage_c_b @ F @ A5 ) @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) ) )
       => ( member_c_b @ F @ ( sigma_measurable_c_b @ M @ N ) ) ) ) ).

% measurableI
thf(fact_536_measurableI,axiom,
    ! [M: sigma_measure_c,F: c > d,N: sigma_measure_d] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ ( sigma_space_c @ M ) )
         => ( member_d @ ( F @ X3 ) @ ( sigma_space_d @ N ) ) )
     => ( ! [A5: set_d] :
            ( ( member_set_d @ A5 @ ( sigma_sets_d @ N ) )
           => ( member_set_c @ ( inf_inf_set_c @ ( vimage_c_d @ F @ A5 ) @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) ) )
       => ( member_c_d @ F @ ( sigma_measurable_c_d @ M @ N ) ) ) ) ).

% measurableI
thf(fact_537_measurableI,axiom,
    ! [M: sigma_measure_b,F: b > c,N: sigma_measure_c] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ ( sigma_space_b @ M ) )
         => ( member_c @ ( F @ X3 ) @ ( sigma_space_c @ N ) ) )
     => ( ! [A5: set_c] :
            ( ( member_set_c @ A5 @ ( sigma_sets_c @ N ) )
           => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_c @ F @ A5 ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) )
       => ( member_b_c @ F @ ( sigma_measurable_b_c @ M @ N ) ) ) ) ).

% measurableI
thf(fact_538_measurableI,axiom,
    ! [M: sigma_measure_d,F: d > c,N: sigma_measure_c] :
      ( ! [X3: d] :
          ( ( member_d @ X3 @ ( sigma_space_d @ M ) )
         => ( member_c @ ( F @ X3 ) @ ( sigma_space_c @ N ) ) )
     => ( ! [A5: set_c] :
            ( ( member_set_c @ A5 @ ( sigma_sets_c @ N ) )
           => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_c @ F @ A5 ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) )
       => ( member_d_c @ F @ ( sigma_measurable_d_c @ M @ N ) ) ) ) ).

% measurableI
thf(fact_539_measurableI,axiom,
    ! [M: sigma_measure_b,F: b > d,N: sigma_measure_d] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ ( sigma_space_b @ M ) )
         => ( member_d @ ( F @ X3 ) @ ( sigma_space_d @ N ) ) )
     => ( ! [A5: set_d] :
            ( ( member_set_d @ A5 @ ( sigma_sets_d @ N ) )
           => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_d @ F @ A5 ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) )
       => ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) ) ) ) ).

% measurableI
thf(fact_540_measurableI,axiom,
    ! [M: sigma_measure_b,F: b > b,N: sigma_measure_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ ( sigma_space_b @ M ) )
         => ( member_b @ ( F @ X3 ) @ ( sigma_space_b @ N ) ) )
     => ( ! [A5: set_b] :
            ( ( member_set_b @ A5 @ ( sigma_sets_b @ N ) )
           => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_b @ F @ A5 ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) )
       => ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) ) ) ) ).

% measurableI
thf(fact_541_measurableI,axiom,
    ! [M: sigma_measure_d,F: d > b,N: sigma_measure_b] :
      ( ! [X3: d] :
          ( ( member_d @ X3 @ ( sigma_space_d @ M ) )
         => ( member_b @ ( F @ X3 ) @ ( sigma_space_b @ N ) ) )
     => ( ! [A5: set_b] :
            ( ( member_set_b @ A5 @ ( sigma_sets_b @ N ) )
           => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_b @ F @ A5 ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) )
       => ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) ) ) ) ).

% measurableI
thf(fact_542_measurableI,axiom,
    ! [M: sigma_measure_d,F: d > d,N: sigma_measure_d] :
      ( ! [X3: d] :
          ( ( member_d @ X3 @ ( sigma_space_d @ M ) )
         => ( member_d @ ( F @ X3 ) @ ( sigma_space_d @ N ) ) )
     => ( ! [A5: set_d] :
            ( ( member_set_d @ A5 @ ( sigma_sets_d @ N ) )
           => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_d @ F @ A5 ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) )
       => ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) ) ) ) ).

% measurableI
thf(fact_543_measurableI,axiom,
    ! [M: sigma_measure_set_d,F: set_d > c,N: sigma_measure_c] :
      ( ! [X3: set_d] :
          ( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
         => ( member_c @ ( F @ X3 ) @ ( sigma_space_c @ N ) ) )
     => ( ! [A5: set_c] :
            ( ( member_set_c @ A5 @ ( sigma_sets_c @ N ) )
           => ( member_set_set_d @ ( inf_inf_set_set_d @ ( vimage_set_d_c @ F @ A5 ) @ ( sigma_space_set_d @ M ) ) @ ( sigma_sets_set_d @ M ) ) )
       => ( member_set_d_c @ F @ ( sigma_1994698223397181103et_d_c @ M @ N ) ) ) ) ).

% measurableI
thf(fact_544_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_set_d,P: $o] :
      ( member_set_set_d
      @ ( collect_set_d
        @ ^ [X2: set_d] :
            ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
            & P ) )
      @ ( sigma_sets_set_d @ M ) ) ).

% sets.sets_Collect_const
thf(fact_545_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_c,P: $o] :
      ( member_set_c
      @ ( collect_c
        @ ^ [X2: c] :
            ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
            & P ) )
      @ ( sigma_sets_c @ M ) ) ).

% sets.sets_Collect_const
thf(fact_546_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_b_d,P: $o] :
      ( member_set_b_d2
      @ ( collect_b_d
        @ ^ [X2: b > d] :
            ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
            & P ) )
      @ ( sigma_sets_b_d @ M ) ) ).

% sets.sets_Collect_const
thf(fact_547_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_b_b,P: $o] :
      ( member_set_b_b2
      @ ( collect_b_b
        @ ^ [X2: b > b] :
            ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
            & P ) )
      @ ( sigma_sets_b_b @ M ) ) ).

% sets.sets_Collect_const
thf(fact_548_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_d_b,P: $o] :
      ( member_set_d_b
      @ ( collect_d_b
        @ ^ [X2: d > b] :
            ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
            & P ) )
      @ ( sigma_sets_d_b @ M ) ) ).

% sets.sets_Collect_const
thf(fact_549_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_d_d,P: $o] :
      ( member_set_d_d
      @ ( collect_d_d
        @ ^ [X2: d > d] :
            ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
            & P ) )
      @ ( sigma_sets_d_d @ M ) ) ).

% sets.sets_Collect_const
thf(fact_550_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_set_b,P: $o] :
      ( member_set_set_b
      @ ( collect_set_b
        @ ^ [X2: set_b] :
            ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
            & P ) )
      @ ( sigma_sets_set_b @ M ) ) ).

% sets.sets_Collect_const
thf(fact_551_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_b,P: $o] :
      ( member_set_b
      @ ( collect_b
        @ ^ [X2: b] :
            ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
            & P ) )
      @ ( sigma_sets_b @ M ) ) ).

% sets.sets_Collect_const
thf(fact_552_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_d,P: $o] :
      ( member_set_d
      @ ( collect_d
        @ ^ [X2: d] :
            ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
            & P ) )
      @ ( sigma_sets_d @ M ) ) ).

% sets.sets_Collect_const
thf(fact_553_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_set_d,P: set_d > $o,Q: set_d > $o] :
      ( ( member_set_set_d
        @ ( collect_set_d
          @ ^ [X2: set_d] :
              ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_set_d @ M ) )
     => ( ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_set_d @ M ) )
       => ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( sigma_sets_set_d @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_554_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_c,P: c > $o,Q: c > $o] :
      ( ( member_set_c
        @ ( collect_c
          @ ^ [X2: c] :
              ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_c @ M ) )
     => ( ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_c @ M ) )
       => ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( sigma_sets_c @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_555_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_b_d,P: ( b > d ) > $o,Q: ( b > d ) > $o] :
      ( ( member_set_b_d2
        @ ( collect_b_d
          @ ^ [X2: b > d] :
              ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b_d @ M ) )
     => ( ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_b_d @ M ) )
       => ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( sigma_sets_b_d @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_556_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_b_b,P: ( b > b ) > $o,Q: ( b > b ) > $o] :
      ( ( member_set_b_b2
        @ ( collect_b_b
          @ ^ [X2: b > b] :
              ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b_b @ M ) )
     => ( ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_b_b @ M ) )
       => ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( sigma_sets_b_b @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_557_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_d_b,P: ( d > b ) > $o,Q: ( d > b ) > $o] :
      ( ( member_set_d_b
        @ ( collect_d_b
          @ ^ [X2: d > b] :
              ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d_b @ M ) )
     => ( ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_d_b @ M ) )
       => ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( sigma_sets_d_b @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_558_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_d_d,P: ( d > d ) > $o,Q: ( d > d ) > $o] :
      ( ( member_set_d_d
        @ ( collect_d_d
          @ ^ [X2: d > d] :
              ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d_d @ M ) )
     => ( ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_d_d @ M ) )
       => ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( sigma_sets_d_d @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_559_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_set_b,P: set_b > $o,Q: set_b > $o] :
      ( ( member_set_set_b
        @ ( collect_set_b
          @ ^ [X2: set_b] :
              ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_set_b @ M ) )
     => ( ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_set_b @ M ) )
       => ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( sigma_sets_set_b @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_560_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_b @ M ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_561_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_d,P: d > $o,Q: d > $o] :
      ( ( member_set_d
        @ ( collect_d
          @ ^ [X2: d] :
              ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d @ M ) )
     => ( ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_d @ M ) )
       => ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( sigma_sets_d @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_562_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_set_d,P: set_d > $o,Q: set_d > $o] :
      ( ( member_set_set_d
        @ ( collect_set_d
          @ ^ [X2: set_d] :
              ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_set_d @ M ) )
     => ( ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_set_d @ M ) )
       => ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_set_d @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_563_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_c,P: c > $o,Q: c > $o] :
      ( ( member_set_c
        @ ( collect_c
          @ ^ [X2: c] :
              ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_c @ M ) )
     => ( ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_c @ M ) )
       => ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_c @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_564_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_b_d,P: ( b > d ) > $o,Q: ( b > d ) > $o] :
      ( ( member_set_b_d2
        @ ( collect_b_d
          @ ^ [X2: b > d] :
              ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b_d @ M ) )
     => ( ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_b_d @ M ) )
       => ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b_d @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_565_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_b_b,P: ( b > b ) > $o,Q: ( b > b ) > $o] :
      ( ( member_set_b_b2
        @ ( collect_b_b
          @ ^ [X2: b > b] :
              ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b_b @ M ) )
     => ( ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_b_b @ M ) )
       => ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b_b @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_566_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_d_b,P: ( d > b ) > $o,Q: ( d > b ) > $o] :
      ( ( member_set_d_b
        @ ( collect_d_b
          @ ^ [X2: d > b] :
              ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d_b @ M ) )
     => ( ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_d_b @ M ) )
       => ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d_b @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_567_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_d_d,P: ( d > d ) > $o,Q: ( d > d ) > $o] :
      ( ( member_set_d_d
        @ ( collect_d_d
          @ ^ [X2: d > d] :
              ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d_d @ M ) )
     => ( ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_d_d @ M ) )
       => ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d_d @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_568_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_set_b,P: set_b > $o,Q: set_b > $o] :
      ( ( member_set_set_b
        @ ( collect_set_b
          @ ^ [X2: set_b] :
              ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_set_b @ M ) )
     => ( ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_set_b @ M ) )
       => ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_set_b @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_569_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_b @ M ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_570_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_d,P: d > $o,Q: d > $o] :
      ( ( member_set_d
        @ ( collect_d
          @ ^ [X2: d] :
              ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d @ M ) )
     => ( ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_d @ M ) )
       => ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_571_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_set_d,P: set_d > $o] :
      ( ( member_set_set_d
        @ ( collect_set_d
          @ ^ [X2: set_d] :
              ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_set_d @ M ) )
     => ( member_set_set_d
        @ ( collect_set_d
          @ ^ [X2: set_d] :
              ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
              & ~ ( P @ X2 ) ) )
        @ ( sigma_sets_set_d @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_572_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_c,P: c > $o] :
      ( ( member_set_c
        @ ( collect_c
          @ ^ [X2: c] :
              ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_c @ M ) )
     => ( member_set_c
        @ ( collect_c
          @ ^ [X2: c] :
              ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
              & ~ ( P @ X2 ) ) )
        @ ( sigma_sets_c @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_573_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_b_d,P: ( b > d ) > $o] :
      ( ( member_set_b_d2
        @ ( collect_b_d
          @ ^ [X2: b > d] :
              ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b_d @ M ) )
     => ( member_set_b_d2
        @ ( collect_b_d
          @ ^ [X2: b > d] :
              ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
              & ~ ( P @ X2 ) ) )
        @ ( sigma_sets_b_d @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_574_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_b_b,P: ( b > b ) > $o] :
      ( ( member_set_b_b2
        @ ( collect_b_b
          @ ^ [X2: b > b] :
              ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b_b @ M ) )
     => ( member_set_b_b2
        @ ( collect_b_b
          @ ^ [X2: b > b] :
              ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
              & ~ ( P @ X2 ) ) )
        @ ( sigma_sets_b_b @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_575_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_d_b,P: ( d > b ) > $o] :
      ( ( member_set_d_b
        @ ( collect_d_b
          @ ^ [X2: d > b] :
              ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d_b @ M ) )
     => ( member_set_d_b
        @ ( collect_d_b
          @ ^ [X2: d > b] :
              ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
              & ~ ( P @ X2 ) ) )
        @ ( sigma_sets_d_b @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_576_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_d_d,P: ( d > d ) > $o] :
      ( ( member_set_d_d
        @ ( collect_d_d
          @ ^ [X2: d > d] :
              ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d_d @ M ) )
     => ( member_set_d_d
        @ ( collect_d_d
          @ ^ [X2: d > d] :
              ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
              & ~ ( P @ X2 ) ) )
        @ ( sigma_sets_d_d @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_577_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_set_b,P: set_b > $o] :
      ( ( member_set_set_b
        @ ( collect_set_b
          @ ^ [X2: set_b] :
              ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_set_b @ M ) )
     => ( member_set_set_b
        @ ( collect_set_b
          @ ^ [X2: set_b] :
              ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
              & ~ ( P @ X2 ) ) )
        @ ( sigma_sets_set_b @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_578_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_b,P: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b @ M ) )
     => ( member_set_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
              & ~ ( P @ X2 ) ) )
        @ ( sigma_sets_b @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_579_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_d,P: d > $o] :
      ( ( member_set_d
        @ ( collect_d
          @ ^ [X2: d] :
              ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d @ M ) )
     => ( member_set_d
        @ ( collect_d
          @ ^ [X2: d] :
              ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
              & ~ ( P @ X2 ) ) )
        @ ( sigma_sets_d @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_580_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_set_d,P: set_d > $o,Q: set_d > $o] :
      ( ( member_set_set_d
        @ ( collect_set_d
          @ ^ [X2: set_d] :
              ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_set_d @ M ) )
     => ( ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_set_d @ M ) )
       => ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( ( Q @ X2 )
                 => ( P @ X2 ) ) ) )
          @ ( sigma_sets_set_d @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_581_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_c,P: c > $o,Q: c > $o] :
      ( ( member_set_c
        @ ( collect_c
          @ ^ [X2: c] :
              ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_c @ M ) )
     => ( ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_c @ M ) )
       => ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( ( Q @ X2 )
                 => ( P @ X2 ) ) ) )
          @ ( sigma_sets_c @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_582_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_b_d,P: ( b > d ) > $o,Q: ( b > d ) > $o] :
      ( ( member_set_b_d2
        @ ( collect_b_d
          @ ^ [X2: b > d] :
              ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b_d @ M ) )
     => ( ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_b_d @ M ) )
       => ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( ( Q @ X2 )
                 => ( P @ X2 ) ) ) )
          @ ( sigma_sets_b_d @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_583_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_b_b,P: ( b > b ) > $o,Q: ( b > b ) > $o] :
      ( ( member_set_b_b2
        @ ( collect_b_b
          @ ^ [X2: b > b] :
              ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b_b @ M ) )
     => ( ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_b_b @ M ) )
       => ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( ( Q @ X2 )
                 => ( P @ X2 ) ) ) )
          @ ( sigma_sets_b_b @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_584_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_d_b,P: ( d > b ) > $o,Q: ( d > b ) > $o] :
      ( ( member_set_d_b
        @ ( collect_d_b
          @ ^ [X2: d > b] :
              ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d_b @ M ) )
     => ( ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_d_b @ M ) )
       => ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( ( Q @ X2 )
                 => ( P @ X2 ) ) ) )
          @ ( sigma_sets_d_b @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_585_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_d_d,P: ( d > d ) > $o,Q: ( d > d ) > $o] :
      ( ( member_set_d_d
        @ ( collect_d_d
          @ ^ [X2: d > d] :
              ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d_d @ M ) )
     => ( ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_d_d @ M ) )
       => ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( ( Q @ X2 )
                 => ( P @ X2 ) ) ) )
          @ ( sigma_sets_d_d @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_586_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_set_b,P: set_b > $o,Q: set_b > $o] :
      ( ( member_set_set_b
        @ ( collect_set_b
          @ ^ [X2: set_b] :
              ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_set_b @ M ) )
     => ( ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_set_b @ M ) )
       => ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( ( Q @ X2 )
                 => ( P @ X2 ) ) ) )
          @ ( sigma_sets_set_b @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_587_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_b @ M ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( ( Q @ X2 )
                 => ( P @ X2 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_588_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_d,P: d > $o,Q: d > $o] :
      ( ( member_set_d
        @ ( collect_d
          @ ^ [X2: d] :
              ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d @ M ) )
     => ( ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( sigma_sets_d @ M ) )
       => ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( ( Q @ X2 )
                 => ( P @ X2 ) ) ) )
          @ ( sigma_sets_d @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_589_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_set_d,Pb: $o] :
      ( member_set_set_d
      @ ( collect_set_d
        @ ^ [X2: set_d] :
            ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
            & Pb ) )
      @ ( sigma_sets_set_d @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_590_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_c,Pb: $o] :
      ( member_set_c
      @ ( collect_c
        @ ^ [X2: c] :
            ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
            & Pb ) )
      @ ( sigma_sets_c @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_591_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_b_d,Pb: $o] :
      ( member_set_b_d2
      @ ( collect_b_d
        @ ^ [X2: b > d] :
            ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
            & Pb ) )
      @ ( sigma_sets_b_d @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_592_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_b_b,Pb: $o] :
      ( member_set_b_b2
      @ ( collect_b_b
        @ ^ [X2: b > b] :
            ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
            & Pb ) )
      @ ( sigma_sets_b_b @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_593_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_d_b,Pb: $o] :
      ( member_set_d_b
      @ ( collect_d_b
        @ ^ [X2: d > b] :
            ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
            & Pb ) )
      @ ( sigma_sets_d_b @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_594_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_d_d,Pb: $o] :
      ( member_set_d_d
      @ ( collect_d_d
        @ ^ [X2: d > d] :
            ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
            & Pb ) )
      @ ( sigma_sets_d_d @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_595_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_set_b,Pb: $o] :
      ( member_set_set_b
      @ ( collect_set_b
        @ ^ [X2: set_b] :
            ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
            & Pb ) )
      @ ( sigma_sets_set_b @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_596_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_b,Pb: $o] :
      ( member_set_b
      @ ( collect_b
        @ ^ [X2: b] :
            ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
            & Pb ) )
      @ ( sigma_sets_b @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_597_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_d,Pb: $o] :
      ( member_set_d
      @ ( collect_d
        @ ^ [X2: d] :
            ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
            & Pb ) )
      @ ( sigma_sets_d @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_598_sets_Osets__into__space,axiom,
    ! [X: set_d,M: sigma_measure_d] :
      ( ( member_set_d @ X @ ( sigma_sets_d @ M ) )
     => ( ord_less_eq_set_d @ X @ ( sigma_space_d @ M ) ) ) ).

% sets.sets_into_space
thf(fact_599_sets_Osets__into__space,axiom,
    ! [X: set_set_b,M: sigma_measure_set_b] :
      ( ( member_set_set_b @ X @ ( sigma_sets_set_b @ M ) )
     => ( ord_le3795704787696855135_set_b @ X @ ( sigma_space_set_b @ M ) ) ) ).

% sets.sets_into_space
thf(fact_600_sets_Osets__into__space,axiom,
    ! [X: set_b,M: sigma_measure_b] :
      ( ( member_set_b @ X @ ( sigma_sets_b @ M ) )
     => ( ord_less_eq_set_b @ X @ ( sigma_space_b @ M ) ) ) ).

% sets.sets_into_space
thf(fact_601_measurable__mono,axiom,
    ! [N2: sigma_measure_d,N: sigma_measure_d,M: sigma_measure_d,M2: sigma_measure_d] :
      ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ N2 ) @ ( sigma_sets_d @ N ) )
     => ( ( ( sigma_space_d @ N )
          = ( sigma_space_d @ N2 ) )
       => ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M ) @ ( sigma_sets_d @ M2 ) )
         => ( ( ( sigma_space_d @ M )
              = ( sigma_space_d @ M2 ) )
           => ( ord_less_eq_set_d_d @ ( sigma_measurable_d_d @ M @ N ) @ ( sigma_measurable_d_d @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_602_measurable__mono,axiom,
    ! [N2: sigma_measure_d,N: sigma_measure_d,M: sigma_measure_b,M2: sigma_measure_b] :
      ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ N2 ) @ ( sigma_sets_d @ N ) )
     => ( ( ( sigma_space_d @ N )
          = ( sigma_space_d @ N2 ) )
       => ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M ) @ ( sigma_sets_b @ M2 ) )
         => ( ( ( sigma_space_b @ M )
              = ( sigma_space_b @ M2 ) )
           => ( ord_less_eq_set_b_d @ ( sigma_measurable_b_d @ M @ N ) @ ( sigma_measurable_b_d @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_603_measurable__mono,axiom,
    ! [N2: sigma_measure_b,N: sigma_measure_b,M: sigma_measure_d,M2: sigma_measure_d] :
      ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ N2 ) @ ( sigma_sets_b @ N ) )
     => ( ( ( sigma_space_b @ N )
          = ( sigma_space_b @ N2 ) )
       => ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M ) @ ( sigma_sets_d @ M2 ) )
         => ( ( ( sigma_space_d @ M )
              = ( sigma_space_d @ M2 ) )
           => ( ord_less_eq_set_d_b @ ( sigma_measurable_d_b @ M @ N ) @ ( sigma_measurable_d_b @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_604_measurable__mono,axiom,
    ! [N2: sigma_measure_b,N: sigma_measure_b,M: sigma_measure_b,M2: sigma_measure_b] :
      ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ N2 ) @ ( sigma_sets_b @ N ) )
     => ( ( ( sigma_space_b @ N )
          = ( sigma_space_b @ N2 ) )
       => ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M ) @ ( sigma_sets_b @ M2 ) )
         => ( ( ( sigma_space_b @ M )
              = ( sigma_space_b @ M2 ) )
           => ( ord_less_eq_set_b_b @ ( sigma_measurable_b_b @ M @ N ) @ ( sigma_measurable_b_b @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_605_le__measureD2,axiom,
    ! [A: sigma_measure_d,B: sigma_measure_d] :
      ( ( ord_le254669808495466590sure_d @ A @ B )
     => ( ( ( sigma_space_d @ A )
          = ( sigma_space_d @ B ) )
       => ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ A ) @ ( sigma_sets_d @ B ) ) ) ) ).

% le_measureD2
thf(fact_606_le__measureD2,axiom,
    ! [A: sigma_measure_b,B: sigma_measure_b] :
      ( ( ord_le254669799889008988sure_b @ A @ B )
     => ( ( ( sigma_space_b @ A )
          = ( sigma_space_b @ B ) )
       => ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ A ) @ ( sigma_sets_b @ B ) ) ) ) ).

% le_measureD2
thf(fact_607_dual__order_Orefl,axiom,
    ! [A2: set_set_b] : ( ord_le3795704787696855135_set_b @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_608_dual__order_Orefl,axiom,
    ! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_609_order__refl,axiom,
    ! [X: set_set_b] : ( ord_le3795704787696855135_set_b @ X @ X ) ).

% order_refl
thf(fact_610_order__refl,axiom,
    ! [X: set_b] : ( ord_less_eq_set_b @ X @ X ) ).

% order_refl
thf(fact_611_le__measureD1,axiom,
    ! [A: sigma_measure_d,B: sigma_measure_d] :
      ( ( ord_le254669808495466590sure_d @ A @ B )
     => ( ord_less_eq_set_d @ ( sigma_space_d @ A ) @ ( sigma_space_d @ B ) ) ) ).

% le_measureD1
thf(fact_612_le__measureD1,axiom,
    ! [A: sigma_measure_set_b,B: sigma_measure_set_b] :
      ( ( ord_le5713619651007674940_set_b @ A @ B )
     => ( ord_le3795704787696855135_set_b @ ( sigma_space_set_b @ A ) @ ( sigma_space_set_b @ B ) ) ) ).

% le_measureD1
thf(fact_613_le__measureD1,axiom,
    ! [A: sigma_measure_b,B: sigma_measure_b] :
      ( ( ord_le254669799889008988sure_b @ A @ B )
     => ( ord_less_eq_set_b @ ( sigma_space_b @ A ) @ ( sigma_space_b @ B ) ) ) ).

% le_measureD1
thf(fact_614_sets__image__in__sets,axiom,
    ! [N: sigma_measure_d,X4: set_d,F: d > b,M: sigma_measure_b] :
      ( ( ( sigma_space_d @ N )
        = X4 )
     => ( ( member_d_b @ F @ ( sigma_measurable_d_b @ N @ M ) )
       => ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ M ) ) @ ( sigma_sets_d @ N ) ) ) ) ).

% sets_image_in_sets
thf(fact_615_sets__image__in__sets,axiom,
    ! [N: sigma_measure_d,X4: set_d,F: d > d,M: sigma_measure_d] :
      ( ( ( sigma_space_d @ N )
        = X4 )
     => ( ( member_d_d @ F @ ( sigma_measurable_d_d @ N @ M ) )
       => ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ M ) ) @ ( sigma_sets_d @ N ) ) ) ) ).

% sets_image_in_sets
thf(fact_616_sets__image__in__sets,axiom,
    ! [N: sigma_measure_b,X4: set_b,F: b > d,M: sigma_measure_d] :
      ( ( ( sigma_space_b @ N )
        = X4 )
     => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ N @ M ) )
       => ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ M ) ) @ ( sigma_sets_b @ N ) ) ) ) ).

% sets_image_in_sets
thf(fact_617_sets__image__in__sets,axiom,
    ! [N: sigma_measure_b,X4: set_b,F: b > b,M: sigma_measure_b] :
      ( ( ( sigma_space_b @ N )
        = X4 )
     => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ N @ M ) )
       => ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ M ) ) @ ( sigma_sets_b @ N ) ) ) ) ).

% sets_image_in_sets
thf(fact_618_inf__Int__eq,axiom,
    ! [R: set_set_d,S: set_set_d] :
      ( ( inf_inf_set_d_o
        @ ^ [X2: set_d] : ( member_set_d @ X2 @ R )
        @ ^ [X2: set_d] : ( member_set_d @ X2 @ S ) )
      = ( ^ [X2: set_d] : ( member_set_d @ X2 @ ( inf_inf_set_set_d @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_619_inf__Int__eq,axiom,
    ! [R: set_c,S: set_c] :
      ( ( inf_inf_c_o
        @ ^ [X2: c] : ( member_c @ X2 @ R )
        @ ^ [X2: c] : ( member_c @ X2 @ S ) )
      = ( ^ [X2: c] : ( member_c @ X2 @ ( inf_inf_set_c @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_620_inf__Int__eq,axiom,
    ! [R: set_b_d,S: set_b_d] :
      ( ( inf_inf_b_d_o
        @ ^ [X2: b > d] : ( member_b_d @ X2 @ R )
        @ ^ [X2: b > d] : ( member_b_d @ X2 @ S ) )
      = ( ^ [X2: b > d] : ( member_b_d @ X2 @ ( inf_inf_set_b_d @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_621_inf__Int__eq,axiom,
    ! [R: set_b_b,S: set_b_b] :
      ( ( inf_inf_b_b_o
        @ ^ [X2: b > b] : ( member_b_b @ X2 @ R )
        @ ^ [X2: b > b] : ( member_b_b @ X2 @ S ) )
      = ( ^ [X2: b > b] : ( member_b_b @ X2 @ ( inf_inf_set_b_b @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_622_inf__Int__eq,axiom,
    ! [R: set_d_b,S: set_d_b] :
      ( ( inf_inf_d_b_o
        @ ^ [X2: d > b] : ( member_d_b @ X2 @ R )
        @ ^ [X2: d > b] : ( member_d_b @ X2 @ S ) )
      = ( ^ [X2: d > b] : ( member_d_b @ X2 @ ( inf_inf_set_d_b @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_623_inf__Int__eq,axiom,
    ! [R: set_d_d,S: set_d_d] :
      ( ( inf_inf_d_d_o
        @ ^ [X2: d > d] : ( member_d_d @ X2 @ R )
        @ ^ [X2: d > d] : ( member_d_d @ X2 @ S ) )
      = ( ^ [X2: d > d] : ( member_d_d @ X2 @ ( inf_inf_set_d_d @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_624_inf__Int__eq,axiom,
    ! [R: set_b,S: set_b] :
      ( ( inf_inf_b_o
        @ ^ [X2: b] : ( member_b @ X2 @ R )
        @ ^ [X2: b] : ( member_b @ X2 @ S ) )
      = ( ^ [X2: b] : ( member_b @ X2 @ ( inf_inf_set_b @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_625_inf__Int__eq,axiom,
    ! [R: set_d,S: set_d] :
      ( ( inf_inf_d_o
        @ ^ [X2: d] : ( member_d @ X2 @ R )
        @ ^ [X2: d] : ( member_d @ X2 @ S ) )
      = ( ^ [X2: d] : ( member_d @ X2 @ ( inf_inf_set_d @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_626_inf__Int__eq,axiom,
    ! [R: set_set_b,S: set_set_b] :
      ( ( inf_inf_set_b_o
        @ ^ [X2: set_b] : ( member_set_b @ X2 @ R )
        @ ^ [X2: set_b] : ( member_set_b @ X2 @ S ) )
      = ( ^ [X2: set_b] : ( member_set_b @ X2 @ ( inf_inf_set_set_b @ R @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_627_pred__subset__eq,axiom,
    ! [R: set_set_d,S: set_set_d] :
      ( ( ord_less_eq_set_d_o
        @ ^ [X2: set_d] : ( member_set_d @ X2 @ R )
        @ ^ [X2: set_d] : ( member_set_d @ X2 @ S ) )
      = ( ord_le3937772867789547105_set_d @ R @ S ) ) ).

% pred_subset_eq
thf(fact_628_pred__subset__eq,axiom,
    ! [R: set_c,S: set_c] :
      ( ( ord_less_eq_c_o
        @ ^ [X2: c] : ( member_c @ X2 @ R )
        @ ^ [X2: c] : ( member_c @ X2 @ S ) )
      = ( ord_less_eq_set_c @ R @ S ) ) ).

% pred_subset_eq
thf(fact_629_pred__subset__eq,axiom,
    ! [R: set_b_d,S: set_b_d] :
      ( ( ord_less_eq_b_d_o
        @ ^ [X2: b > d] : ( member_b_d @ X2 @ R )
        @ ^ [X2: b > d] : ( member_b_d @ X2 @ S ) )
      = ( ord_less_eq_set_b_d @ R @ S ) ) ).

% pred_subset_eq
thf(fact_630_pred__subset__eq,axiom,
    ! [R: set_b_b,S: set_b_b] :
      ( ( ord_less_eq_b_b_o
        @ ^ [X2: b > b] : ( member_b_b @ X2 @ R )
        @ ^ [X2: b > b] : ( member_b_b @ X2 @ S ) )
      = ( ord_less_eq_set_b_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_631_pred__subset__eq,axiom,
    ! [R: set_d_b,S: set_d_b] :
      ( ( ord_less_eq_d_b_o
        @ ^ [X2: d > b] : ( member_d_b @ X2 @ R )
        @ ^ [X2: d > b] : ( member_d_b @ X2 @ S ) )
      = ( ord_less_eq_set_d_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_632_pred__subset__eq,axiom,
    ! [R: set_d_d,S: set_d_d] :
      ( ( ord_less_eq_d_d_o
        @ ^ [X2: d > d] : ( member_d_d @ X2 @ R )
        @ ^ [X2: d > d] : ( member_d_d @ X2 @ S ) )
      = ( ord_less_eq_set_d_d @ R @ S ) ) ).

% pred_subset_eq
thf(fact_633_pred__subset__eq,axiom,
    ! [R: set_set_b,S: set_set_b] :
      ( ( ord_less_eq_set_b_o
        @ ^ [X2: set_b] : ( member_set_b @ X2 @ R )
        @ ^ [X2: set_b] : ( member_set_b @ X2 @ S ) )
      = ( ord_le3795704787696855135_set_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_634_pred__subset__eq,axiom,
    ! [R: set_b,S: set_b] :
      ( ( ord_less_eq_b_o
        @ ^ [X2: b] : ( member_b @ X2 @ R )
        @ ^ [X2: b] : ( member_b @ X2 @ S ) )
      = ( ord_less_eq_set_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_635_subset__Collect__iff,axiom,
    ! [B: set_set_d,A: set_set_d,P: set_d > $o] :
      ( ( ord_le3937772867789547105_set_d @ B @ A )
     => ( ( ord_le3937772867789547105_set_d @ B
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ A )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: set_d] :
              ( ( member_set_d @ X2 @ B )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_636_subset__Collect__iff,axiom,
    ! [B: set_c,A: set_c,P: c > $o] :
      ( ( ord_less_eq_set_c @ B @ A )
     => ( ( ord_less_eq_set_c @ B
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ A )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: c] :
              ( ( member_c @ X2 @ B )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_637_subset__Collect__iff,axiom,
    ! [B: set_b_d,A: set_b_d,P: ( b > d ) > $o] :
      ( ( ord_less_eq_set_b_d @ B @ A )
     => ( ( ord_less_eq_set_b_d @ B
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ A )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: b > d] :
              ( ( member_b_d @ X2 @ B )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_638_subset__Collect__iff,axiom,
    ! [B: set_b_b,A: set_b_b,P: ( b > b ) > $o] :
      ( ( ord_less_eq_set_b_b @ B @ A )
     => ( ( ord_less_eq_set_b_b @ B
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ A )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: b > b] :
              ( ( member_b_b @ X2 @ B )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_639_subset__Collect__iff,axiom,
    ! [B: set_d_b,A: set_d_b,P: ( d > b ) > $o] :
      ( ( ord_less_eq_set_d_b @ B @ A )
     => ( ( ord_less_eq_set_d_b @ B
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ A )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: d > b] :
              ( ( member_d_b @ X2 @ B )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_640_subset__Collect__iff,axiom,
    ! [B: set_d_d,A: set_d_d,P: ( d > d ) > $o] :
      ( ( ord_less_eq_set_d_d @ B @ A )
     => ( ( ord_less_eq_set_d_d @ B
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ A )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: d > d] :
              ( ( member_d_d @ X2 @ B )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_641_subset__Collect__iff,axiom,
    ! [B: set_set_b,A: set_set_b,P: set_b > $o] :
      ( ( ord_le3795704787696855135_set_b @ B @ A )
     => ( ( ord_le3795704787696855135_set_b @ B
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ A )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: set_b] :
              ( ( member_set_b @ X2 @ B )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_642_subset__Collect__iff,axiom,
    ! [B: set_b,A: set_b,P: b > $o] :
      ( ( ord_less_eq_set_b @ B @ A )
     => ( ( ord_less_eq_set_b @ B
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ A )
                & ( P @ X2 ) ) ) )
        = ( ! [X2: b] :
              ( ( member_b @ X2 @ B )
             => ( P @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_643_subset__CollectI,axiom,
    ! [B: set_set_d,A: set_set_d,Q: set_d > $o,P: set_d > $o] :
      ( ( ord_le3937772867789547105_set_d @ B @ A )
     => ( ! [X3: set_d] :
            ( ( member_set_d @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le3937772867789547105_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ B )
                & ( Q @ X2 ) ) )
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ A )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_644_subset__CollectI,axiom,
    ! [B: set_c,A: set_c,Q: c > $o,P: c > $o] :
      ( ( ord_less_eq_set_c @ B @ A )
     => ( ! [X3: c] :
            ( ( member_c @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ B )
                & ( Q @ X2 ) ) )
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ A )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_645_subset__CollectI,axiom,
    ! [B: set_b_d,A: set_b_d,Q: ( b > d ) > $o,P: ( b > d ) > $o] :
      ( ( ord_less_eq_set_b_d @ B @ A )
     => ( ! [X3: b > d] :
            ( ( member_b_d @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_b_d
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ B )
                & ( Q @ X2 ) ) )
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ A )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_646_subset__CollectI,axiom,
    ! [B: set_b_b,A: set_b_b,Q: ( b > b ) > $o,P: ( b > b ) > $o] :
      ( ( ord_less_eq_set_b_b @ B @ A )
     => ( ! [X3: b > b] :
            ( ( member_b_b @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_b_b
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ B )
                & ( Q @ X2 ) ) )
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ A )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_647_subset__CollectI,axiom,
    ! [B: set_d_b,A: set_d_b,Q: ( d > b ) > $o,P: ( d > b ) > $o] :
      ( ( ord_less_eq_set_d_b @ B @ A )
     => ( ! [X3: d > b] :
            ( ( member_d_b @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ B )
                & ( Q @ X2 ) ) )
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ A )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_648_subset__CollectI,axiom,
    ! [B: set_d_d,A: set_d_d,Q: ( d > d ) > $o,P: ( d > d ) > $o] :
      ( ( ord_less_eq_set_d_d @ B @ A )
     => ( ! [X3: d > d] :
            ( ( member_d_d @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ B )
                & ( Q @ X2 ) ) )
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ A )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_649_subset__CollectI,axiom,
    ! [B: set_set_b,A: set_set_b,Q: set_b > $o,P: set_b > $o] :
      ( ( ord_le3795704787696855135_set_b @ B @ A )
     => ( ! [X3: set_b] :
            ( ( member_set_b @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le3795704787696855135_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ B )
                & ( Q @ X2 ) ) )
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ A )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_650_subset__CollectI,axiom,
    ! [B: set_b,A: set_b,Q: b > $o,P: b > $o] :
      ( ( ord_less_eq_set_b @ B @ A )
     => ( ! [X3: b] :
            ( ( member_b @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ B )
                & ( Q @ X2 ) ) )
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ A )
                & ( P @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_651_Collect__restrict,axiom,
    ! [X4: set_set_d,P: set_d > $o] :
      ( ord_le3937772867789547105_set_d
      @ ( collect_set_d
        @ ^ [X2: set_d] :
            ( ( member_set_d @ X2 @ X4 )
            & ( P @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_652_Collect__restrict,axiom,
    ! [X4: set_c,P: c > $o] :
      ( ord_less_eq_set_c
      @ ( collect_c
        @ ^ [X2: c] :
            ( ( member_c @ X2 @ X4 )
            & ( P @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_653_Collect__restrict,axiom,
    ! [X4: set_b_d,P: ( b > d ) > $o] :
      ( ord_less_eq_set_b_d
      @ ( collect_b_d
        @ ^ [X2: b > d] :
            ( ( member_b_d @ X2 @ X4 )
            & ( P @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_654_Collect__restrict,axiom,
    ! [X4: set_b_b,P: ( b > b ) > $o] :
      ( ord_less_eq_set_b_b
      @ ( collect_b_b
        @ ^ [X2: b > b] :
            ( ( member_b_b @ X2 @ X4 )
            & ( P @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_655_Collect__restrict,axiom,
    ! [X4: set_d_b,P: ( d > b ) > $o] :
      ( ord_less_eq_set_d_b
      @ ( collect_d_b
        @ ^ [X2: d > b] :
            ( ( member_d_b @ X2 @ X4 )
            & ( P @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_656_Collect__restrict,axiom,
    ! [X4: set_d_d,P: ( d > d ) > $o] :
      ( ord_less_eq_set_d_d
      @ ( collect_d_d
        @ ^ [X2: d > d] :
            ( ( member_d_d @ X2 @ X4 )
            & ( P @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_657_Collect__restrict,axiom,
    ! [X4: set_set_b,P: set_b > $o] :
      ( ord_le3795704787696855135_set_b
      @ ( collect_set_b
        @ ^ [X2: set_b] :
            ( ( member_set_b @ X2 @ X4 )
            & ( P @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_658_Collect__restrict,axiom,
    ! [X4: set_b,P: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X2: b] :
            ( ( member_b @ X2 @ X4 )
            & ( P @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_659_prop__restrict,axiom,
    ! [X: set_d,Z4: set_set_d,X4: set_set_d,P: set_d > $o] :
      ( ( member_set_d @ X @ Z4 )
     => ( ( ord_le3937772867789547105_set_d @ Z4
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ X4 )
                & ( P @ X2 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_660_prop__restrict,axiom,
    ! [X: c,Z4: set_c,X4: set_c,P: c > $o] :
      ( ( member_c @ X @ Z4 )
     => ( ( ord_less_eq_set_c @ Z4
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ X4 )
                & ( P @ X2 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_661_prop__restrict,axiom,
    ! [X: b > d,Z4: set_b_d,X4: set_b_d,P: ( b > d ) > $o] :
      ( ( member_b_d @ X @ Z4 )
     => ( ( ord_less_eq_set_b_d @ Z4
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ X4 )
                & ( P @ X2 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_662_prop__restrict,axiom,
    ! [X: b > b,Z4: set_b_b,X4: set_b_b,P: ( b > b ) > $o] :
      ( ( member_b_b @ X @ Z4 )
     => ( ( ord_less_eq_set_b_b @ Z4
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ X4 )
                & ( P @ X2 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_663_prop__restrict,axiom,
    ! [X: d > b,Z4: set_d_b,X4: set_d_b,P: ( d > b ) > $o] :
      ( ( member_d_b @ X @ Z4 )
     => ( ( ord_less_eq_set_d_b @ Z4
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ X4 )
                & ( P @ X2 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_664_prop__restrict,axiom,
    ! [X: d > d,Z4: set_d_d,X4: set_d_d,P: ( d > d ) > $o] :
      ( ( member_d_d @ X @ Z4 )
     => ( ( ord_less_eq_set_d_d @ Z4
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ X4 )
                & ( P @ X2 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_665_prop__restrict,axiom,
    ! [X: set_b,Z4: set_set_b,X4: set_set_b,P: set_b > $o] :
      ( ( member_set_b @ X @ Z4 )
     => ( ( ord_le3795704787696855135_set_b @ Z4
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ X4 )
                & ( P @ X2 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_666_prop__restrict,axiom,
    ! [X: b,Z4: set_b,X4: set_b,P: b > $o] :
      ( ( member_b @ X @ Z4 )
     => ( ( ord_less_eq_set_b @ Z4
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ X4 )
                & ( P @ X2 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_667_vimage__algebra__cong,axiom,
    ! [X4: set_c,Y2: set_c,F: c > d,G: c > d,M: sigma_measure_d,N: sigma_measure_d] :
      ( ( X4 = Y2 )
     => ( ! [X3: c] :
            ( ( member_c @ X3 @ Y2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( ( sigma_sets_d @ M )
            = ( sigma_sets_d @ N ) )
         => ( ( sigma_5666859195879803797ra_c_d @ X4 @ F @ M )
            = ( sigma_5666859195879803797ra_c_d @ Y2 @ G @ N ) ) ) ) ) ).

% vimage_algebra_cong
thf(fact_668_vimage__algebra__cong,axiom,
    ! [X4: set_c,Y2: set_c,F: c > b,G: c > b,M: sigma_measure_b,N: sigma_measure_b] :
      ( ( X4 = Y2 )
     => ( ! [X3: c] :
            ( ( member_c @ X3 @ Y2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( ( sigma_sets_b @ M )
            = ( sigma_sets_b @ N ) )
         => ( ( sigma_5666859195879803795ra_c_b @ X4 @ F @ M )
            = ( sigma_5666859195879803795ra_c_b @ Y2 @ G @ N ) ) ) ) ) ).

% vimage_algebra_cong
thf(fact_669_vimage__algebra__cong,axiom,
    ! [X4: set_set_d,Y2: set_set_d,F: set_d > d,G: set_d > d,M: sigma_measure_d,N: sigma_measure_d] :
      ( ( X4 = Y2 )
     => ( ! [X3: set_d] :
            ( ( member_set_d @ X3 @ Y2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( ( sigma_sets_d @ M )
            = ( sigma_sets_d @ N ) )
         => ( ( sigma_5482745410756188660et_d_d @ X4 @ F @ M )
            = ( sigma_5482745410756188660et_d_d @ Y2 @ G @ N ) ) ) ) ) ).

% vimage_algebra_cong
thf(fact_670_vimage__algebra__cong,axiom,
    ! [X4: set_set_b,Y2: set_set_b,F: set_b > d,G: set_b > d,M: sigma_measure_d,N: sigma_measure_d] :
      ( ( X4 = Y2 )
     => ( ! [X3: set_b] :
            ( ( member_set_b @ X3 @ Y2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( ( sigma_sets_d @ M )
            = ( sigma_sets_d @ N ) )
         => ( ( sigma_6754043411966162038et_b_d @ X4 @ F @ M )
            = ( sigma_6754043411966162038et_b_d @ Y2 @ G @ N ) ) ) ) ) ).

% vimage_algebra_cong
thf(fact_671_vimage__algebra__cong,axiom,
    ! [X4: set_set_d,Y2: set_set_d,F: set_d > b,G: set_d > b,M: sigma_measure_b,N: sigma_measure_b] :
      ( ( X4 = Y2 )
     => ( ! [X3: set_d] :
            ( ( member_set_d @ X3 @ Y2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( ( sigma_sets_b @ M )
            = ( sigma_sets_b @ N ) )
         => ( ( sigma_5482745410756188658et_d_b @ X4 @ F @ M )
            = ( sigma_5482745410756188658et_d_b @ Y2 @ G @ N ) ) ) ) ) ).

% vimage_algebra_cong
thf(fact_672_vimage__algebra__cong,axiom,
    ! [X4: set_set_b,Y2: set_set_b,F: set_b > b,G: set_b > b,M: sigma_measure_b,N: sigma_measure_b] :
      ( ( X4 = Y2 )
     => ( ! [X3: set_b] :
            ( ( member_set_b @ X3 @ Y2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( ( sigma_sets_b @ M )
            = ( sigma_sets_b @ N ) )
         => ( ( sigma_6754043411966162036et_b_b @ X4 @ F @ M )
            = ( sigma_6754043411966162036et_b_b @ Y2 @ G @ N ) ) ) ) ) ).

% vimage_algebra_cong
thf(fact_673_vimage__algebra__cong,axiom,
    ! [X4: set_b_d,Y2: set_b_d,F: ( b > d ) > d,G: ( b > d ) > d,M: sigma_measure_d,N: sigma_measure_d] :
      ( ( X4 = Y2 )
     => ( ! [X3: b > d] :
            ( ( member_b_d @ X3 @ Y2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( ( sigma_sets_d @ M )
            = ( sigma_sets_d @ N ) )
         => ( ( sigma_8883386097941561860_b_d_d @ X4 @ F @ M )
            = ( sigma_8883386097941561860_b_d_d @ Y2 @ G @ N ) ) ) ) ) ).

% vimage_algebra_cong
thf(fact_674_vimage__algebra__cong,axiom,
    ! [X4: set_b_b,Y2: set_b_b,F: ( b > b ) > d,G: ( b > b ) > d,M: sigma_measure_d,N: sigma_measure_d] :
      ( ( X4 = Y2 )
     => ( ! [X3: b > b] :
            ( ( member_b_b @ X3 @ Y2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( ( sigma_sets_d @ M )
            = ( sigma_sets_d @ N ) )
         => ( ( sigma_931312062296759430_b_b_d @ X4 @ F @ M )
            = ( sigma_931312062296759430_b_b_d @ Y2 @ G @ N ) ) ) ) ) ).

% vimage_algebra_cong
thf(fact_675_vimage__algebra__cong,axiom,
    ! [X4: set_d_b,Y2: set_d_b,F: ( d > b ) > d,G: ( d > b ) > d,M: sigma_measure_d,N: sigma_measure_d] :
      ( ( X4 = Y2 )
     => ( ! [X3: d > b] :
            ( ( member_d_b @ X3 @ Y2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( ( sigma_sets_d @ M )
            = ( sigma_sets_d @ N ) )
         => ( ( sigma_2900377255818245000_d_b_d @ X4 @ F @ M )
            = ( sigma_2900377255818245000_d_b_d @ Y2 @ G @ N ) ) ) ) ) ).

% vimage_algebra_cong
thf(fact_676_vimage__algebra__cong,axiom,
    ! [X4: set_d_d,Y2: set_d_d,F: ( d > d ) > d,G: ( d > d ) > d,M: sigma_measure_d,N: sigma_measure_d] :
      ( ( X4 = Y2 )
     => ( ! [X3: d > d] :
            ( ( member_d_d @ X3 @ Y2 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( ( sigma_sets_d @ M )
            = ( sigma_sets_d @ N ) )
         => ( ( sigma_1629079254608271622_d_d_d @ X4 @ F @ M )
            = ( sigma_1629079254608271622_d_d_d @ Y2 @ G @ N ) ) ) ) ) ).

% vimage_algebra_cong
thf(fact_677_sets__vimage__algebra__cong,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d,X4: set_d,F: d > d] :
      ( ( ( sigma_sets_d @ M )
        = ( sigma_sets_d @ N ) )
     => ( ( sigma_sets_d @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ M ) )
        = ( sigma_sets_d @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ N ) ) ) ) ).

% sets_vimage_algebra_cong
thf(fact_678_sets__vimage__algebra__cong,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d,X4: set_b,F: b > d] :
      ( ( ( sigma_sets_d @ M )
        = ( sigma_sets_d @ N ) )
     => ( ( sigma_sets_b @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ M ) )
        = ( sigma_sets_b @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ N ) ) ) ) ).

% sets_vimage_algebra_cong
thf(fact_679_sets__vimage__algebra__cong,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b,X4: set_d,F: d > b] :
      ( ( ( sigma_sets_b @ M )
        = ( sigma_sets_b @ N ) )
     => ( ( sigma_sets_d @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ M ) )
        = ( sigma_sets_d @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ N ) ) ) ) ).

% sets_vimage_algebra_cong
thf(fact_680_sets__vimage__algebra__cong,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b,X4: set_b,F: b > b] :
      ( ( ( sigma_sets_b @ M )
        = ( sigma_sets_b @ N ) )
     => ( ( sigma_sets_b @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ M ) )
        = ( sigma_sets_b @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ N ) ) ) ) ).

% sets_vimage_algebra_cong
thf(fact_681_mono__vimage__algebra,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d,X4: set_d,F: d > d] :
      ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M ) @ ( sigma_sets_d @ N ) )
     => ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ M ) ) @ ( sigma_sets_d @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ N ) ) ) ) ).

% mono_vimage_algebra
thf(fact_682_mono__vimage__algebra,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d,X4: set_b,F: b > d] :
      ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M ) @ ( sigma_sets_d @ N ) )
     => ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ M ) ) @ ( sigma_sets_b @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ N ) ) ) ) ).

% mono_vimage_algebra
thf(fact_683_mono__vimage__algebra,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b,X4: set_d,F: d > b] :
      ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M ) @ ( sigma_sets_b @ N ) )
     => ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ M ) ) @ ( sigma_sets_d @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ N ) ) ) ) ).

% mono_vimage_algebra
thf(fact_684_mono__vimage__algebra,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b,X4: set_b,F: b > b] :
      ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M ) @ ( sigma_sets_b @ N ) )
     => ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ M ) ) @ ( sigma_sets_b @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ N ) ) ) ) ).

% mono_vimage_algebra
thf(fact_685_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_set_b,Z2: set_set_b] : ( Y4 = Z2 ) )
    = ( ^ [X2: set_set_b,Y3: set_set_b] :
          ( ( ord_le3795704787696855135_set_b @ X2 @ Y3 )
          & ( ord_le3795704787696855135_set_b @ Y3 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_686_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_b,Z2: set_b] : ( Y4 = Z2 ) )
    = ( ^ [X2: set_b,Y3: set_b] :
          ( ( ord_less_eq_set_b @ X2 @ Y3 )
          & ( ord_less_eq_set_b @ Y3 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_687_ord__eq__le__trans,axiom,
    ! [A2: set_set_b,B2: set_set_b,C2: set_set_b] :
      ( ( A2 = B2 )
     => ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
       => ( ord_le3795704787696855135_set_b @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_688_ord__eq__le__trans,axiom,
    ! [A2: set_b,B2: set_b,C2: set_b] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_b @ B2 @ C2 )
       => ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_689_ord__le__eq__trans,axiom,
    ! [A2: set_set_b,B2: set_set_b,C2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_le3795704787696855135_set_b @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_690_ord__le__eq__trans,axiom,
    ! [A2: set_b,B2: set_b,C2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_691_order__antisym,axiom,
    ! [X: set_set_b,Y: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ X @ Y )
     => ( ( ord_le3795704787696855135_set_b @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_692_order__antisym,axiom,
    ! [X: set_b,Y: set_b] :
      ( ( ord_less_eq_set_b @ X @ Y )
     => ( ( ord_less_eq_set_b @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_693_order_Otrans,axiom,
    ! [A2: set_set_b,B2: set_set_b,C2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
       => ( ord_le3795704787696855135_set_b @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_694_order_Otrans,axiom,
    ! [A2: set_b,B2: set_b,C2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( ( ord_less_eq_set_b @ B2 @ C2 )
       => ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_695_order__trans,axiom,
    ! [X: set_set_b,Y: set_set_b,Z: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ X @ Y )
     => ( ( ord_le3795704787696855135_set_b @ Y @ Z )
       => ( ord_le3795704787696855135_set_b @ X @ Z ) ) ) ).

% order_trans
thf(fact_696_order__trans,axiom,
    ! [X: set_b,Y: set_b,Z: set_b] :
      ( ( ord_less_eq_set_b @ X @ Y )
     => ( ( ord_less_eq_set_b @ Y @ Z )
       => ( ord_less_eq_set_b @ X @ Z ) ) ) ).

% order_trans
thf(fact_697_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_set_b,Z2: set_set_b] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_set_b,B4: set_set_b] :
          ( ( ord_le3795704787696855135_set_b @ B4 @ A4 )
          & ( ord_le3795704787696855135_set_b @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_698_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_b,Z2: set_b] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_b,B4: set_b] :
          ( ( ord_less_eq_set_b @ B4 @ A4 )
          & ( ord_less_eq_set_b @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_699_dual__order_Oantisym,axiom,
    ! [B2: set_set_b,A2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ B2 @ A2 )
     => ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_700_dual__order_Oantisym,axiom,
    ! [B2: set_b,A2: set_b] :
      ( ( ord_less_eq_set_b @ B2 @ A2 )
     => ( ( ord_less_eq_set_b @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_701_dual__order_Otrans,axiom,
    ! [B2: set_set_b,A2: set_set_b,C2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ B2 @ A2 )
     => ( ( ord_le3795704787696855135_set_b @ C2 @ B2 )
       => ( ord_le3795704787696855135_set_b @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_702_dual__order_Otrans,axiom,
    ! [B2: set_b,A2: set_b,C2: set_b] :
      ( ( ord_less_eq_set_b @ B2 @ A2 )
     => ( ( ord_less_eq_set_b @ C2 @ B2 )
       => ( ord_less_eq_set_b @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_703_antisym,axiom,
    ! [A2: set_set_b,B2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( ( ord_le3795704787696855135_set_b @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_704_antisym,axiom,
    ! [A2: set_b,B2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( ( ord_less_eq_set_b @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_705_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_set_b,Z2: set_set_b] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_set_b,B4: set_set_b] :
          ( ( ord_le3795704787696855135_set_b @ A4 @ B4 )
          & ( ord_le3795704787696855135_set_b @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_706_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_b,Z2: set_b] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_b,B4: set_b] :
          ( ( ord_less_eq_set_b @ A4 @ B4 )
          & ( ord_less_eq_set_b @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_707_order__subst1,axiom,
    ! [A2: set_set_b,F: set_set_b > set_set_b,B2: set_set_b,C2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
       => ( ! [X3: set_set_b,Y5: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ X3 @ Y5 )
             => ( ord_le3795704787696855135_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3795704787696855135_set_b @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_708_order__subst1,axiom,
    ! [A2: set_set_b,F: set_b > set_set_b,B2: set_b,C2: set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_b @ B2 @ C2 )
       => ( ! [X3: set_b,Y5: set_b] :
              ( ( ord_less_eq_set_b @ X3 @ Y5 )
             => ( ord_le3795704787696855135_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3795704787696855135_set_b @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_709_order__subst1,axiom,
    ! [A2: set_b,F: set_set_b > set_b,B2: set_set_b,C2: set_set_b] :
      ( ( ord_less_eq_set_b @ A2 @ ( F @ B2 ) )
     => ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
       => ( ! [X3: set_set_b,Y5: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ X3 @ Y5 )
             => ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_b @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_710_order__subst1,axiom,
    ! [A2: set_b,F: set_b > set_b,B2: set_b,C2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_set_b @ B2 @ C2 )
       => ( ! [X3: set_b,Y5: set_b] :
              ( ( ord_less_eq_set_b @ X3 @ Y5 )
             => ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_b @ A2 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_711_order__subst2,axiom,
    ! [A2: set_set_b,B2: set_set_b,F: set_set_b > set_set_b,C2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( ( ord_le3795704787696855135_set_b @ ( F @ B2 ) @ C2 )
       => ( ! [X3: set_set_b,Y5: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ X3 @ Y5 )
             => ( ord_le3795704787696855135_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3795704787696855135_set_b @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_712_order__subst2,axiom,
    ! [A2: set_set_b,B2: set_set_b,F: set_set_b > set_b,C2: set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( ( ord_less_eq_set_b @ ( F @ B2 ) @ C2 )
       => ( ! [X3: set_set_b,Y5: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ X3 @ Y5 )
             => ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_b @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_713_order__subst2,axiom,
    ! [A2: set_b,B2: set_b,F: set_b > set_set_b,C2: set_set_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( ( ord_le3795704787696855135_set_b @ ( F @ B2 ) @ C2 )
       => ( ! [X3: set_b,Y5: set_b] :
              ( ( ord_less_eq_set_b @ X3 @ Y5 )
             => ( ord_le3795704787696855135_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3795704787696855135_set_b @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_714_order__subst2,axiom,
    ! [A2: set_b,B2: set_b,F: set_b > set_b,C2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( ( ord_less_eq_set_b @ ( F @ B2 ) @ C2 )
       => ( ! [X3: set_b,Y5: set_b] :
              ( ( ord_less_eq_set_b @ X3 @ Y5 )
             => ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_b @ ( F @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_715_order__eq__refl,axiom,
    ! [X: set_set_b,Y: set_set_b] :
      ( ( X = Y )
     => ( ord_le3795704787696855135_set_b @ X @ Y ) ) ).

% order_eq_refl
thf(fact_716_order__eq__refl,axiom,
    ! [X: set_b,Y: set_b] :
      ( ( X = Y )
     => ( ord_less_eq_set_b @ X @ Y ) ) ).

% order_eq_refl
thf(fact_717_ord__eq__le__subst,axiom,
    ! [A2: set_set_b,F: set_set_b > set_set_b,B2: set_set_b,C2: set_set_b] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
       => ( ! [X3: set_set_b,Y5: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ X3 @ Y5 )
             => ( ord_le3795704787696855135_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3795704787696855135_set_b @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_718_ord__eq__le__subst,axiom,
    ! [A2: set_b,F: set_set_b > set_b,B2: set_set_b,C2: set_set_b] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_le3795704787696855135_set_b @ B2 @ C2 )
       => ( ! [X3: set_set_b,Y5: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ X3 @ Y5 )
             => ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_b @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_719_ord__eq__le__subst,axiom,
    ! [A2: set_set_b,F: set_b > set_set_b,B2: set_b,C2: set_b] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_b @ B2 @ C2 )
       => ( ! [X3: set_b,Y5: set_b] :
              ( ( ord_less_eq_set_b @ X3 @ Y5 )
             => ( ord_le3795704787696855135_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3795704787696855135_set_b @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_720_ord__eq__le__subst,axiom,
    ! [A2: set_b,F: set_b > set_b,B2: set_b,C2: set_b] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_set_b @ B2 @ C2 )
       => ( ! [X3: set_b,Y5: set_b] :
              ( ( ord_less_eq_set_b @ X3 @ Y5 )
             => ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_b @ A2 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_721_ord__le__eq__subst,axiom,
    ! [A2: set_set_b,B2: set_set_b,F: set_set_b > set_set_b,C2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: set_set_b,Y5: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ X3 @ Y5 )
             => ( ord_le3795704787696855135_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3795704787696855135_set_b @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_722_ord__le__eq__subst,axiom,
    ! [A2: set_set_b,B2: set_set_b,F: set_set_b > set_b,C2: set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: set_set_b,Y5: set_set_b] :
              ( ( ord_le3795704787696855135_set_b @ X3 @ Y5 )
             => ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_b @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_723_ord__le__eq__subst,axiom,
    ! [A2: set_b,B2: set_b,F: set_b > set_set_b,C2: set_set_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: set_b,Y5: set_b] :
              ( ( ord_less_eq_set_b @ X3 @ Y5 )
             => ( ord_le3795704787696855135_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_le3795704787696855135_set_b @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_724_ord__le__eq__subst,axiom,
    ! [A2: set_b,B2: set_b,F: set_b > set_b,C2: set_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X3: set_b,Y5: set_b] :
              ( ( ord_less_eq_set_b @ X3 @ Y5 )
             => ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_set_b @ ( F @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_725_order__antisym__conv,axiom,
    ! [Y: set_set_b,X: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ Y @ X )
     => ( ( ord_le3795704787696855135_set_b @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_726_order__antisym__conv,axiom,
    ! [Y: set_b,X: set_b] :
      ( ( ord_less_eq_set_b @ Y @ X )
     => ( ( ord_less_eq_set_b @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_727_sets__eq__iff__bounded,axiom,
    ! [A: sigma_measure_d,B: sigma_measure_d,C: sigma_measure_d] :
      ( ( ord_le254669808495466590sure_d @ A @ B )
     => ( ( ord_le254669808495466590sure_d @ B @ C )
       => ( ( ( sigma_sets_d @ A )
            = ( sigma_sets_d @ C ) )
         => ( ( sigma_sets_d @ B )
            = ( sigma_sets_d @ A ) ) ) ) ) ).

% sets_eq_iff_bounded
thf(fact_728_sets__eq__iff__bounded,axiom,
    ! [A: sigma_measure_b,B: sigma_measure_b,C: sigma_measure_b] :
      ( ( ord_le254669799889008988sure_b @ A @ B )
     => ( ( ord_le254669799889008988sure_b @ B @ C )
       => ( ( ( sigma_sets_b @ A )
            = ( sigma_sets_b @ C ) )
         => ( ( sigma_sets_b @ B )
            = ( sigma_sets_b @ A ) ) ) ) ) ).

% sets_eq_iff_bounded
thf(fact_729_sets__image__in__sets_H,axiom,
    ! [X4: set_d,N: sigma_measure_d,M: sigma_measure_d,F: d > d] :
      ( ( member_set_d @ X4 @ ( sigma_sets_d @ N ) )
     => ( ! [A5: set_d] :
            ( ( member_set_d @ A5 @ ( sigma_sets_d @ M ) )
           => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_d @ F @ A5 ) @ X4 ) @ ( sigma_sets_d @ N ) ) )
       => ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ M ) ) @ ( sigma_sets_d @ N ) ) ) ) ).

% sets_image_in_sets'
thf(fact_730_sets__image__in__sets_H,axiom,
    ! [X4: set_d,N: sigma_measure_d,M: sigma_measure_b,F: d > b] :
      ( ( member_set_d @ X4 @ ( sigma_sets_d @ N ) )
     => ( ! [A5: set_b] :
            ( ( member_set_b @ A5 @ ( sigma_sets_b @ M ) )
           => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_b @ F @ A5 ) @ X4 ) @ ( sigma_sets_d @ N ) ) )
       => ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ M ) ) @ ( sigma_sets_d @ N ) ) ) ) ).

% sets_image_in_sets'
thf(fact_731_sets__image__in__sets_H,axiom,
    ! [X4: set_set_b,N: sigma_measure_set_b,M: sigma_measure_d,F: set_b > d] :
      ( ( member_set_set_b @ X4 @ ( sigma_sets_set_b @ N ) )
     => ( ! [A5: set_d] :
            ( ( member_set_d @ A5 @ ( sigma_sets_d @ M ) )
           => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ A5 ) @ X4 ) @ ( sigma_sets_set_b @ N ) ) )
       => ( ord_le3201067847557142847_set_b @ ( sigma_sets_set_b @ ( sigma_6754043411966162038et_b_d @ X4 @ F @ M ) ) @ ( sigma_sets_set_b @ N ) ) ) ) ).

% sets_image_in_sets'
thf(fact_732_sets__image__in__sets_H,axiom,
    ! [X4: set_set_b,N: sigma_measure_set_b,M: sigma_measure_b,F: set_b > b] :
      ( ( member_set_set_b @ X4 @ ( sigma_sets_set_b @ N ) )
     => ( ! [A5: set_b] :
            ( ( member_set_b @ A5 @ ( sigma_sets_b @ M ) )
           => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ A5 ) @ X4 ) @ ( sigma_sets_set_b @ N ) ) )
       => ( ord_le3201067847557142847_set_b @ ( sigma_sets_set_b @ ( sigma_6754043411966162036et_b_b @ X4 @ F @ M ) ) @ ( sigma_sets_set_b @ N ) ) ) ) ).

% sets_image_in_sets'
thf(fact_733_sets__image__in__sets_H,axiom,
    ! [X4: set_b,N: sigma_measure_b,M: sigma_measure_d,F: b > d] :
      ( ( member_set_b @ X4 @ ( sigma_sets_b @ N ) )
     => ( ! [A5: set_d] :
            ( ( member_set_d @ A5 @ ( sigma_sets_d @ M ) )
           => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_d @ F @ A5 ) @ X4 ) @ ( sigma_sets_b @ N ) ) )
       => ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ M ) ) @ ( sigma_sets_b @ N ) ) ) ) ).

% sets_image_in_sets'
thf(fact_734_sets__image__in__sets_H,axiom,
    ! [X4: set_b,N: sigma_measure_b,M: sigma_measure_b,F: b > b] :
      ( ( member_set_b @ X4 @ ( sigma_sets_b @ N ) )
     => ( ! [A5: set_b] :
            ( ( member_set_b @ A5 @ ( sigma_sets_b @ M ) )
           => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_b @ F @ A5 ) @ X4 ) @ ( sigma_sets_b @ N ) ) )
       => ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ M ) ) @ ( sigma_sets_b @ N ) ) ) ) ).

% sets_image_in_sets'
thf(fact_735_in__vimage__algebra,axiom,
    ! [A: set_d,M: sigma_measure_d,F: b > d,X4: set_b] :
      ( ( member_set_d @ A @ ( sigma_sets_d @ M ) )
     => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_d @ F @ A ) @ X4 ) @ ( sigma_sets_b @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ M ) ) ) ) ).

% in_vimage_algebra
thf(fact_736_in__vimage__algebra,axiom,
    ! [A: set_b,M: sigma_measure_b,F: b > b,X4: set_b] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ M ) )
     => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_b @ F @ A ) @ X4 ) @ ( sigma_sets_b @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ M ) ) ) ) ).

% in_vimage_algebra
thf(fact_737_in__vimage__algebra,axiom,
    ! [A: set_d,M: sigma_measure_d,F: d > d,X4: set_d] :
      ( ( member_set_d @ A @ ( sigma_sets_d @ M ) )
     => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_d @ F @ A ) @ X4 ) @ ( sigma_sets_d @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ M ) ) ) ) ).

% in_vimage_algebra
thf(fact_738_in__vimage__algebra,axiom,
    ! [A: set_b,M: sigma_measure_b,F: d > b,X4: set_d] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ M ) )
     => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_b @ F @ A ) @ X4 ) @ ( sigma_sets_d @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ M ) ) ) ) ).

% in_vimage_algebra
thf(fact_739_in__vimage__algebra,axiom,
    ! [A: set_d,M: sigma_measure_d,F: set_b > d,X4: set_set_b] :
      ( ( member_set_d @ A @ ( sigma_sets_d @ M ) )
     => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ A ) @ X4 ) @ ( sigma_sets_set_b @ ( sigma_6754043411966162038et_b_d @ X4 @ F @ M ) ) ) ) ).

% in_vimage_algebra
thf(fact_740_in__vimage__algebra,axiom,
    ! [A: set_b,M: sigma_measure_b,F: set_b > b,X4: set_set_b] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ M ) )
     => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ A ) @ X4 ) @ ( sigma_sets_set_b @ ( sigma_6754043411966162036et_b_b @ X4 @ F @ M ) ) ) ) ).

% in_vimage_algebra
thf(fact_741_increasingD,axiom,
    ! [M: set_set_d,F: set_d > set_set_b,X: set_d,Y: set_d] :
      ( ( measur7802506524085621681_set_b @ M @ F )
     => ( ( ord_less_eq_set_d @ X @ Y )
       => ( ( member_set_d @ X @ M )
         => ( ( member_set_d @ Y @ M )
           => ( ord_le3795704787696855135_set_b @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_742_increasingD,axiom,
    ! [M: set_set_d,F: set_d > set_b,X: set_d,Y: set_d] :
      ( ( measur2652592430956242641_set_b @ M @ F )
     => ( ( ord_less_eq_set_d @ X @ Y )
       => ( ( member_set_d @ X @ M )
         => ( ( member_set_d @ Y @ M )
           => ( ord_less_eq_set_b @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_743_increasingD,axiom,
    ! [M: set_set_set_b,F: set_set_b > set_set_b,X: set_set_b,Y: set_set_b] :
      ( ( measur8864879078030831187_set_b @ M @ F )
     => ( ( ord_le3795704787696855135_set_b @ X @ Y )
       => ( ( member_set_set_b @ X @ M )
         => ( ( member_set_set_b @ Y @ M )
           => ( ord_le3795704787696855135_set_b @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_744_increasingD,axiom,
    ! [M: set_set_set_b,F: set_set_b > set_b,X: set_set_b,Y: set_set_b] :
      ( ( measur6668990532724476275_set_b @ M @ F )
     => ( ( ord_le3795704787696855135_set_b @ X @ Y )
       => ( ( member_set_set_b @ X @ M )
         => ( ( member_set_set_b @ Y @ M )
           => ( ord_less_eq_set_b @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_745_increasingD,axiom,
    ! [M: set_set_b,F: set_b > set_set_b,X: set_b,Y: set_b] :
      ( ( measur1967322967037640499_set_b @ M @ F )
     => ( ( ord_less_eq_set_b @ X @ Y )
       => ( ( member_set_b @ X @ M )
         => ( ( member_set_b @ Y @ M )
           => ( ord_le3795704787696855135_set_b @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_746_increasingD,axiom,
    ! [M: set_set_b,F: set_b > set_b,X: set_b,Y: set_b] :
      ( ( measur9187034394192042579_set_b @ M @ F )
     => ( ( ord_less_eq_set_b @ X @ Y )
       => ( ( member_set_b @ X @ M )
         => ( ( member_set_b @ Y @ M )
           => ( ord_less_eq_set_b @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_747_increasing__def,axiom,
    ( measur8864879078030831187_set_b
    = ( ^ [M3: set_set_set_b,Mu: set_set_b > set_set_b] :
        ! [X2: set_set_b] :
          ( ( member_set_set_b @ X2 @ M3 )
         => ! [Y3: set_set_b] :
              ( ( member_set_set_b @ Y3 @ M3 )
             => ( ( ord_le3795704787696855135_set_b @ X2 @ Y3 )
               => ( ord_le3795704787696855135_set_b @ ( Mu @ X2 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).

% increasing_def
thf(fact_748_increasing__def,axiom,
    ( measur6668990532724476275_set_b
    = ( ^ [M3: set_set_set_b,Mu: set_set_b > set_b] :
        ! [X2: set_set_b] :
          ( ( member_set_set_b @ X2 @ M3 )
         => ! [Y3: set_set_b] :
              ( ( member_set_set_b @ Y3 @ M3 )
             => ( ( ord_le3795704787696855135_set_b @ X2 @ Y3 )
               => ( ord_less_eq_set_b @ ( Mu @ X2 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).

% increasing_def
thf(fact_749_increasing__def,axiom,
    ( measur1967322967037640499_set_b
    = ( ^ [M3: set_set_b,Mu: set_b > set_set_b] :
        ! [X2: set_b] :
          ( ( member_set_b @ X2 @ M3 )
         => ! [Y3: set_b] :
              ( ( member_set_b @ Y3 @ M3 )
             => ( ( ord_less_eq_set_b @ X2 @ Y3 )
               => ( ord_le3795704787696855135_set_b @ ( Mu @ X2 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).

% increasing_def
thf(fact_750_increasing__def,axiom,
    ( measur9187034394192042579_set_b
    = ( ^ [M3: set_set_b,Mu: set_b > set_b] :
        ! [X2: set_b] :
          ( ( member_set_b @ X2 @ M3 )
         => ! [Y3: set_b] :
              ( ( member_set_b @ Y3 @ M3 )
             => ( ( ord_less_eq_set_b @ X2 @ Y3 )
               => ( ord_less_eq_set_b @ ( Mu @ X2 ) @ ( Mu @ Y3 ) ) ) ) ) ) ) ).

% increasing_def
thf(fact_751_space__sup__measure_H,axiom,
    ! [B: sigma_measure_b,A: sigma_measure_b] :
      ( ( ( sigma_sets_b @ B )
        = ( sigma_sets_b @ A ) )
     => ( ( sigma_space_b @ ( measur3004909623614618065sure_b @ A @ B ) )
        = ( sigma_space_b @ A ) ) ) ).

% space_sup_measure'
thf(fact_752_space__sup__measure_H,axiom,
    ! [B: sigma_measure_d,A: sigma_measure_d] :
      ( ( ( sigma_sets_d @ B )
        = ( sigma_sets_d @ A ) )
     => ( ( sigma_space_d @ ( measur3004909623614618067sure_d @ A @ B ) )
        = ( sigma_space_d @ A ) ) ) ).

% space_sup_measure'
thf(fact_753_sets__vimage__algebra2,axiom,
    ! [F: b > b,X4: set_b,M: sigma_measure_b] :
      ( ( member_b_b @ F
        @ ( pi_b_b @ X4
          @ ^ [Uu: b] : ( sigma_space_b @ M ) ) )
     => ( ( sigma_sets_b @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ M ) )
        = ( collect_set_b
          @ ^ [Uu: set_b] :
            ? [A3: set_b] :
              ( ( Uu
                = ( inf_inf_set_b @ ( vimage_b_b @ F @ A3 ) @ X4 ) )
              & ( member_set_b @ A3 @ ( sigma_sets_b @ M ) ) ) ) ) ) ).

% sets_vimage_algebra2
thf(fact_754_sets__vimage__algebra2,axiom,
    ! [F: b > d,X4: set_b,M: sigma_measure_d] :
      ( ( member_b_d @ F
        @ ( pi_b_d @ X4
          @ ^ [Uu: b] : ( sigma_space_d @ M ) ) )
     => ( ( sigma_sets_b @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ M ) )
        = ( collect_set_b
          @ ^ [Uu: set_b] :
            ? [A3: set_d] :
              ( ( Uu
                = ( inf_inf_set_b @ ( vimage_b_d @ F @ A3 ) @ X4 ) )
              & ( member_set_d @ A3 @ ( sigma_sets_d @ M ) ) ) ) ) ) ).

% sets_vimage_algebra2
thf(fact_755_sets__vimage__algebra2,axiom,
    ! [F: d > b,X4: set_d,M: sigma_measure_b] :
      ( ( member_d_b @ F
        @ ( pi_d_b @ X4
          @ ^ [Uu: d] : ( sigma_space_b @ M ) ) )
     => ( ( sigma_sets_d @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ M ) )
        = ( collect_set_d
          @ ^ [Uu: set_d] :
            ? [A3: set_b] :
              ( ( Uu
                = ( inf_inf_set_d @ ( vimage_d_b @ F @ A3 ) @ X4 ) )
              & ( member_set_b @ A3 @ ( sigma_sets_b @ M ) ) ) ) ) ) ).

% sets_vimage_algebra2
thf(fact_756_sets__vimage__algebra2,axiom,
    ! [F: d > d,X4: set_d,M: sigma_measure_d] :
      ( ( member_d_d @ F
        @ ( pi_d_d @ X4
          @ ^ [Uu: d] : ( sigma_space_d @ M ) ) )
     => ( ( sigma_sets_d @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ M ) )
        = ( collect_set_d
          @ ^ [Uu: set_d] :
            ? [A3: set_d] :
              ( ( Uu
                = ( inf_inf_set_d @ ( vimage_d_d @ F @ A3 ) @ X4 ) )
              & ( member_set_d @ A3 @ ( sigma_sets_d @ M ) ) ) ) ) ) ).

% sets_vimage_algebra2
thf(fact_757_sets__vimage__algebra2,axiom,
    ! [F: set_b > b,X4: set_set_b,M: sigma_measure_b] :
      ( ( member_set_b_b @ F
        @ ( pi_set_b_b @ X4
          @ ^ [Uu: set_b] : ( sigma_space_b @ M ) ) )
     => ( ( sigma_sets_set_b @ ( sigma_6754043411966162036et_b_b @ X4 @ F @ M ) )
        = ( collect_set_set_b
          @ ^ [Uu: set_set_b] :
            ? [A3: set_b] :
              ( ( Uu
                = ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ A3 ) @ X4 ) )
              & ( member_set_b @ A3 @ ( sigma_sets_b @ M ) ) ) ) ) ) ).

% sets_vimage_algebra2
thf(fact_758_sets__vimage__algebra2,axiom,
    ! [F: set_b > d,X4: set_set_b,M: sigma_measure_d] :
      ( ( member_set_b_d @ F
        @ ( pi_set_b_d @ X4
          @ ^ [Uu: set_b] : ( sigma_space_d @ M ) ) )
     => ( ( sigma_sets_set_b @ ( sigma_6754043411966162038et_b_d @ X4 @ F @ M ) )
        = ( collect_set_set_b
          @ ^ [Uu: set_set_b] :
            ? [A3: set_d] :
              ( ( Uu
                = ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ A3 ) @ X4 ) )
              & ( member_set_d @ A3 @ ( sigma_sets_d @ M ) ) ) ) ) ) ).

% sets_vimage_algebra2
thf(fact_759_sets__vimage__algebra,axiom,
    ! [X4: set_b,F: b > d,M: sigma_measure_d] :
      ( ( sigma_sets_b @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ M ) )
      = ( sigma_sigma_sets_b @ X4
        @ ( collect_set_b
          @ ^ [Uu: set_b] :
            ? [A3: set_d] :
              ( ( Uu
                = ( inf_inf_set_b @ ( vimage_b_d @ F @ A3 ) @ X4 ) )
              & ( member_set_d @ A3 @ ( sigma_sets_d @ M ) ) ) ) ) ) ).

% sets_vimage_algebra
thf(fact_760_sets__vimage__algebra,axiom,
    ! [X4: set_b,F: b > b,M: sigma_measure_b] :
      ( ( sigma_sets_b @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ M ) )
      = ( sigma_sigma_sets_b @ X4
        @ ( collect_set_b
          @ ^ [Uu: set_b] :
            ? [A3: set_b] :
              ( ( Uu
                = ( inf_inf_set_b @ ( vimage_b_b @ F @ A3 ) @ X4 ) )
              & ( member_set_b @ A3 @ ( sigma_sets_b @ M ) ) ) ) ) ) ).

% sets_vimage_algebra
thf(fact_761_sets__vimage__algebra,axiom,
    ! [X4: set_d,F: d > d,M: sigma_measure_d] :
      ( ( sigma_sets_d @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ M ) )
      = ( sigma_sigma_sets_d @ X4
        @ ( collect_set_d
          @ ^ [Uu: set_d] :
            ? [A3: set_d] :
              ( ( Uu
                = ( inf_inf_set_d @ ( vimage_d_d @ F @ A3 ) @ X4 ) )
              & ( member_set_d @ A3 @ ( sigma_sets_d @ M ) ) ) ) ) ) ).

% sets_vimage_algebra
thf(fact_762_sets__vimage__algebra,axiom,
    ! [X4: set_d,F: d > b,M: sigma_measure_b] :
      ( ( sigma_sets_d @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ M ) )
      = ( sigma_sigma_sets_d @ X4
        @ ( collect_set_d
          @ ^ [Uu: set_d] :
            ? [A3: set_b] :
              ( ( Uu
                = ( inf_inf_set_d @ ( vimage_d_b @ F @ A3 ) @ X4 ) )
              & ( member_set_b @ A3 @ ( sigma_sets_b @ M ) ) ) ) ) ) ).

% sets_vimage_algebra
thf(fact_763_sets__vimage__algebra,axiom,
    ! [X4: set_set_b,F: set_b > d,M: sigma_measure_d] :
      ( ( sigma_sets_set_b @ ( sigma_6754043411966162038et_b_d @ X4 @ F @ M ) )
      = ( sigma_2987359972167793591_set_b @ X4
        @ ( collect_set_set_b
          @ ^ [Uu: set_set_b] :
            ? [A3: set_d] :
              ( ( Uu
                = ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ A3 ) @ X4 ) )
              & ( member_set_d @ A3 @ ( sigma_sets_d @ M ) ) ) ) ) ) ).

% sets_vimage_algebra
thf(fact_764_sets__vimage__algebra,axiom,
    ! [X4: set_set_b,F: set_b > b,M: sigma_measure_b] :
      ( ( sigma_sets_set_b @ ( sigma_6754043411966162036et_b_b @ X4 @ F @ M ) )
      = ( sigma_2987359972167793591_set_b @ X4
        @ ( collect_set_set_b
          @ ^ [Uu: set_set_b] :
            ? [A3: set_b] :
              ( ( Uu
                = ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ A3 ) @ X4 ) )
              & ( member_set_b @ A3 @ ( sigma_sets_b @ M ) ) ) ) ) ) ).

% sets_vimage_algebra
thf(fact_765_le__measure__iff,axiom,
    ( ord_le254669808495466590sure_d
    = ( ^ [M3: sigma_measure_d,N3: sigma_measure_d] :
          ( ( ( ( sigma_space_d @ M3 )
              = ( sigma_space_d @ N3 ) )
           => ( ( ( ( sigma_sets_d @ M3 )
                  = ( sigma_sets_d @ N3 ) )
               => ( ord_le5694847263963582634nnreal @ ( sigma_emeasure_d @ M3 ) @ ( sigma_emeasure_d @ N3 ) ) )
              & ( ( ( sigma_sets_d @ M3 )
                 != ( sigma_sets_d @ N3 ) )
               => ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M3 ) @ ( sigma_sets_d @ N3 ) ) ) ) )
          & ( ( ( sigma_space_d @ M3 )
             != ( sigma_space_d @ N3 ) )
           => ( ord_less_eq_set_d @ ( sigma_space_d @ M3 ) @ ( sigma_space_d @ N3 ) ) ) ) ) ) ).

% le_measure_iff
thf(fact_766_le__measure__iff,axiom,
    ( ord_le5713619651007674940_set_b
    = ( ^ [M3: sigma_measure_set_b,N3: sigma_measure_set_b] :
          ( ( ( ( sigma_space_set_b @ M3 )
              = ( sigma_space_set_b @ N3 ) )
           => ( ( ( ( sigma_sets_set_b @ M3 )
                  = ( sigma_sets_set_b @ N3 ) )
               => ( ord_le3779468636425607816nnreal @ ( sigma_emeasure_set_b @ M3 ) @ ( sigma_emeasure_set_b @ N3 ) ) )
              & ( ( ( sigma_sets_set_b @ M3 )
                 != ( sigma_sets_set_b @ N3 ) )
               => ( ord_le3201067847557142847_set_b @ ( sigma_sets_set_b @ M3 ) @ ( sigma_sets_set_b @ N3 ) ) ) ) )
          & ( ( ( sigma_space_set_b @ M3 )
             != ( sigma_space_set_b @ N3 ) )
           => ( ord_le3795704787696855135_set_b @ ( sigma_space_set_b @ M3 ) @ ( sigma_space_set_b @ N3 ) ) ) ) ) ) ).

% le_measure_iff
thf(fact_767_le__measure__iff,axiom,
    ( ord_le254669799889008988sure_b
    = ( ^ [M3: sigma_measure_b,N3: sigma_measure_b] :
          ( ( ( ( sigma_space_b @ M3 )
              = ( sigma_space_b @ N3 ) )
           => ( ( ( ( sigma_sets_b @ M3 )
                  = ( sigma_sets_b @ N3 ) )
               => ( ord_le6365330890766322088nnreal @ ( sigma_emeasure_b @ M3 ) @ ( sigma_emeasure_b @ N3 ) ) )
              & ( ( ( sigma_sets_b @ M3 )
                 != ( sigma_sets_b @ N3 ) )
               => ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M3 ) @ ( sigma_sets_b @ N3 ) ) ) ) )
          & ( ( ( sigma_space_b @ M3 )
             != ( sigma_space_b @ N3 ) )
           => ( ord_less_eq_set_b @ ( sigma_space_b @ M3 ) @ ( sigma_space_b @ N3 ) ) ) ) ) ) ).

% le_measure_iff
thf(fact_768_measurable__iff__sets,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
      = ( ( member_d_b @ F
          @ ( pi_d_b @ ( sigma_space_d @ M )
            @ ^ [Uu: d] : ( sigma_space_b @ N ) ) )
        & ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ ( sigma_2879403613907127762ra_d_b @ ( sigma_space_d @ M ) @ F @ N ) ) @ ( sigma_sets_d @ M ) ) ) ) ).

% measurable_iff_sets
thf(fact_769_measurable__iff__sets,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
      = ( ( member_d_d @ F
          @ ( pi_d_d @ ( sigma_space_d @ M )
            @ ^ [Uu: d] : ( sigma_space_d @ N ) ) )
        & ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ ( sigma_2879403613907127764ra_d_d @ ( sigma_space_d @ M ) @ F @ N ) ) @ ( sigma_sets_d @ M ) ) ) ) ).

% measurable_iff_sets
thf(fact_770_measurable__iff__sets,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
      = ( ( member_b_d @ F
          @ ( pi_b_d @ ( sigma_space_b @ M )
            @ ^ [Uu: b] : ( sigma_space_d @ N ) ) )
        & ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ ( sigma_8454314777852479830ra_b_d @ ( sigma_space_b @ M ) @ F @ N ) ) @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_iff_sets
thf(fact_771_measurable__iff__sets,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
      = ( ( member_b_b @ F
          @ ( pi_b_b @ ( sigma_space_b @ M )
            @ ^ [Uu: b] : ( sigma_space_b @ N ) ) )
        & ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ ( sigma_8454314777852479828ra_b_b @ ( sigma_space_b @ M ) @ F @ N ) ) @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_iff_sets
thf(fact_772_conj__subset__def,axiom,
    ! [A: set_set_b,P: set_b > $o,Q: set_b > $o] :
      ( ( ord_le3795704787696855135_set_b @ A
        @ ( collect_set_b
          @ ^ [X2: set_b] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) )
      = ( ( ord_le3795704787696855135_set_b @ A @ ( collect_set_b @ P ) )
        & ( ord_le3795704787696855135_set_b @ A @ ( collect_set_b @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_773_conj__subset__def,axiom,
    ! [A: set_b,P: b > $o,Q: b > $o] :
      ( ( ord_less_eq_set_b @ A
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) )
      = ( ( ord_less_eq_set_b @ A @ ( collect_b @ P ) )
        & ( ord_less_eq_set_b @ A @ ( collect_b @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_774_sets__sup__measure_H,axiom,
    ! [B: sigma_measure_d,A: sigma_measure_d] :
      ( ( ( sigma_sets_d @ B )
        = ( sigma_sets_d @ A ) )
     => ( ( sigma_sets_d @ ( measur3004909623614618067sure_d @ A @ B ) )
        = ( sigma_sets_d @ A ) ) ) ).

% sets_sup_measure'
thf(fact_775_sets__sup__measure_H,axiom,
    ! [B: sigma_measure_b,A: sigma_measure_b] :
      ( ( ( sigma_sets_b @ B )
        = ( sigma_sets_b @ A ) )
     => ( ( sigma_sets_b @ ( measur3004909623614618065sure_b @ A @ B ) )
        = ( sigma_sets_b @ A ) ) ) ).

% sets_sup_measure'
thf(fact_776_sigma__sets__top,axiom,
    ! [Sp: set_d,A: set_set_d] : ( member_set_d @ Sp @ ( sigma_sigma_sets_d @ Sp @ A ) ) ).

% sigma_sets_top
thf(fact_777_sigma__sets__top,axiom,
    ! [Sp: set_b,A: set_set_b] : ( member_set_b @ Sp @ ( sigma_sigma_sets_b @ Sp @ A ) ) ).

% sigma_sets_top
thf(fact_778_sigma__sets__eqI,axiom,
    ! [A: set_set_d,M: set_d,B: set_set_d] :
      ( ! [A6: set_d] :
          ( ( member_set_d @ A6 @ A )
         => ( member_set_d @ A6 @ ( sigma_sigma_sets_d @ M @ B ) ) )
     => ( ! [B5: set_d] :
            ( ( member_set_d @ B5 @ B )
           => ( member_set_d @ B5 @ ( sigma_sigma_sets_d @ M @ A ) ) )
       => ( ( sigma_sigma_sets_d @ M @ A )
          = ( sigma_sigma_sets_d @ M @ B ) ) ) ) ).

% sigma_sets_eqI
thf(fact_779_sigma__sets__eqI,axiom,
    ! [A: set_set_b,M: set_b,B: set_set_b] :
      ( ! [A6: set_b] :
          ( ( member_set_b @ A6 @ A )
         => ( member_set_b @ A6 @ ( sigma_sigma_sets_b @ M @ B ) ) )
     => ( ! [B5: set_b] :
            ( ( member_set_b @ B5 @ B )
           => ( member_set_b @ B5 @ ( sigma_sigma_sets_b @ M @ A ) ) )
       => ( ( sigma_sigma_sets_b @ M @ A )
          = ( sigma_sigma_sets_b @ M @ B ) ) ) ) ).

% sigma_sets_eqI
thf(fact_780_sigma__sets_OBasic,axiom,
    ! [A2: set_d,A: set_set_d,Sp: set_d] :
      ( ( member_set_d @ A2 @ A )
     => ( member_set_d @ A2 @ ( sigma_sigma_sets_d @ Sp @ A ) ) ) ).

% sigma_sets.Basic
thf(fact_781_sigma__sets_OBasic,axiom,
    ! [A2: set_b,A: set_set_b,Sp: set_b] :
      ( ( member_set_b @ A2 @ A )
     => ( member_set_b @ A2 @ ( sigma_sigma_sets_b @ Sp @ A ) ) ) ).

% sigma_sets.Basic
thf(fact_782_le__emeasure__sup__measure_H2,axiom,
    ! [B: sigma_measure_d,A: sigma_measure_d,X4: set_d] :
      ( ( ( sigma_sets_d @ B )
        = ( sigma_sets_d @ A ) )
     => ( ( member_set_d @ X4 @ ( sigma_sets_d @ A ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_d @ B @ X4 ) @ ( sigma_emeasure_d @ ( measur3004909623614618067sure_d @ A @ B ) @ X4 ) ) ) ) ).

% le_emeasure_sup_measure'2
thf(fact_783_le__emeasure__sup__measure_H2,axiom,
    ! [B: sigma_measure_b,A: sigma_measure_b,X4: set_b] :
      ( ( ( sigma_sets_b @ B )
        = ( sigma_sets_b @ A ) )
     => ( ( member_set_b @ X4 @ ( sigma_sets_b @ A ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_b @ B @ X4 ) @ ( sigma_emeasure_b @ ( measur3004909623614618065sure_b @ A @ B ) @ X4 ) ) ) ) ).

% le_emeasure_sup_measure'2
thf(fact_784_le__emeasure__sup__measure_H1,axiom,
    ! [B: sigma_measure_d,A: sigma_measure_d,X4: set_d] :
      ( ( ( sigma_sets_d @ B )
        = ( sigma_sets_d @ A ) )
     => ( ( member_set_d @ X4 @ ( sigma_sets_d @ A ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_d @ A @ X4 ) @ ( sigma_emeasure_d @ ( measur3004909623614618067sure_d @ A @ B ) @ X4 ) ) ) ) ).

% le_emeasure_sup_measure'1
thf(fact_785_le__emeasure__sup__measure_H1,axiom,
    ! [B: sigma_measure_b,A: sigma_measure_b,X4: set_b] :
      ( ( ( sigma_sets_b @ B )
        = ( sigma_sets_b @ A ) )
     => ( ( member_set_b @ X4 @ ( sigma_sets_b @ A ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_b @ A @ X4 ) @ ( sigma_emeasure_b @ ( measur3004909623614618065sure_b @ A @ B ) @ X4 ) ) ) ) ).

% le_emeasure_sup_measure'1
thf(fact_786_sigma__sets__superset__generator,axiom,
    ! [A: set_set_b,X4: set_b] : ( ord_le3795704787696855135_set_b @ A @ ( sigma_sigma_sets_b @ X4 @ A ) ) ).

% sigma_sets_superset_generator
thf(fact_787_sigma__sets__subseteq,axiom,
    ! [A: set_set_b,B: set_set_b,X4: set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ord_le3795704787696855135_set_b @ ( sigma_sigma_sets_b @ X4 @ A ) @ ( sigma_sigma_sets_b @ X4 @ B ) ) ) ).

% sigma_sets_subseteq
thf(fact_788_sigma__sets__mono,axiom,
    ! [A: set_set_b,X4: set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( sigma_sigma_sets_b @ X4 @ B ) )
     => ( ord_le3795704787696855135_set_b @ ( sigma_sigma_sets_b @ X4 @ A ) @ ( sigma_sigma_sets_b @ X4 @ B ) ) ) ).

% sigma_sets_mono
thf(fact_789_measure__eqI,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d] :
      ( ( ( sigma_sets_d @ M )
        = ( sigma_sets_d @ N ) )
     => ( ! [A5: set_d] :
            ( ( member_set_d @ A5 @ ( sigma_sets_d @ M ) )
           => ( ( sigma_emeasure_d @ M @ A5 )
              = ( sigma_emeasure_d @ N @ A5 ) ) )
       => ( M = N ) ) ) ).

% measure_eqI
thf(fact_790_measure__eqI,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b] :
      ( ( ( sigma_sets_b @ M )
        = ( sigma_sets_b @ N ) )
     => ( ! [A5: set_b] :
            ( ( member_set_b @ A5 @ ( sigma_sets_b @ M ) )
           => ( ( sigma_emeasure_b @ M @ A5 )
              = ( sigma_emeasure_b @ N @ A5 ) ) )
       => ( M = N ) ) ) ).

% measure_eqI
thf(fact_791_emeasure__space,axiom,
    ! [M: sigma_measure_b,A: set_b] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_b @ M @ A ) @ ( sigma_emeasure_b @ M @ ( sigma_space_b @ M ) ) ) ).

% emeasure_space
thf(fact_792_emeasure__space,axiom,
    ! [M: sigma_measure_d,A: set_d] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_d @ M @ A ) @ ( sigma_emeasure_d @ M @ ( sigma_space_d @ M ) ) ) ).

% emeasure_space
thf(fact_793_emeasure__sup__measure_H__le2,axiom,
    ! [B: sigma_measure_d,C: sigma_measure_d,A: sigma_measure_d,X4: set_d] :
      ( ( ( sigma_sets_d @ B )
        = ( sigma_sets_d @ C ) )
     => ( ( ( sigma_sets_d @ A )
          = ( sigma_sets_d @ C ) )
       => ( ( member_set_d @ X4 @ ( sigma_sets_d @ C ) )
         => ( ! [Y6: set_d] :
                ( ( ord_less_eq_set_d @ Y6 @ X4 )
               => ( ( member_set_d @ Y6 @ ( sigma_sets_d @ A ) )
                 => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_d @ A @ Y6 ) @ ( sigma_emeasure_d @ C @ Y6 ) ) ) )
           => ( ! [Y6: set_d] :
                  ( ( ord_less_eq_set_d @ Y6 @ X4 )
                 => ( ( member_set_d @ Y6 @ ( sigma_sets_d @ A ) )
                   => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_d @ B @ Y6 ) @ ( sigma_emeasure_d @ C @ Y6 ) ) ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_d @ ( measur3004909623614618067sure_d @ A @ B ) @ X4 ) @ ( sigma_emeasure_d @ C @ X4 ) ) ) ) ) ) ) ).

% emeasure_sup_measure'_le2
thf(fact_794_emeasure__sup__measure_H__le2,axiom,
    ! [B: sigma_measure_set_b,C: sigma_measure_set_b,A: sigma_measure_set_b,X4: set_set_b] :
      ( ( ( sigma_sets_set_b @ B )
        = ( sigma_sets_set_b @ C ) )
     => ( ( ( sigma_sets_set_b @ A )
          = ( sigma_sets_set_b @ C ) )
       => ( ( member_set_set_b @ X4 @ ( sigma_sets_set_b @ C ) )
         => ( ! [Y6: set_set_b] :
                ( ( ord_le3795704787696855135_set_b @ Y6 @ X4 )
               => ( ( member_set_set_b @ Y6 @ ( sigma_sets_set_b @ A ) )
                 => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_set_b @ A @ Y6 ) @ ( sigma_emeasure_set_b @ C @ Y6 ) ) ) )
           => ( ! [Y6: set_set_b] :
                  ( ( ord_le3795704787696855135_set_b @ Y6 @ X4 )
                 => ( ( member_set_set_b @ Y6 @ ( sigma_sets_set_b @ A ) )
                   => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_set_b @ B @ Y6 ) @ ( sigma_emeasure_set_b @ C @ Y6 ) ) ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_set_b @ ( measur132476219272276657_set_b @ A @ B ) @ X4 ) @ ( sigma_emeasure_set_b @ C @ X4 ) ) ) ) ) ) ) ).

% emeasure_sup_measure'_le2
thf(fact_795_emeasure__sup__measure_H__le2,axiom,
    ! [B: sigma_measure_b,C: sigma_measure_b,A: sigma_measure_b,X4: set_b] :
      ( ( ( sigma_sets_b @ B )
        = ( sigma_sets_b @ C ) )
     => ( ( ( sigma_sets_b @ A )
          = ( sigma_sets_b @ C ) )
       => ( ( member_set_b @ X4 @ ( sigma_sets_b @ C ) )
         => ( ! [Y6: set_b] :
                ( ( ord_less_eq_set_b @ Y6 @ X4 )
               => ( ( member_set_b @ Y6 @ ( sigma_sets_b @ A ) )
                 => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_b @ A @ Y6 ) @ ( sigma_emeasure_b @ C @ Y6 ) ) ) )
           => ( ! [Y6: set_b] :
                  ( ( ord_less_eq_set_b @ Y6 @ X4 )
                 => ( ( member_set_b @ Y6 @ ( sigma_sets_b @ A ) )
                   => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_b @ B @ Y6 ) @ ( sigma_emeasure_b @ C @ Y6 ) ) ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_b @ ( measur3004909623614618065sure_b @ A @ B ) @ X4 ) @ ( sigma_emeasure_b @ C @ X4 ) ) ) ) ) ) ) ).

% emeasure_sup_measure'_le2
thf(fact_796_sigma__sets__vimage__commute,axiom,
    ! [X4: b > d,Omega: set_b,Omega2: set_d,M2: set_set_d] :
      ( ( member_b_d @ X4
        @ ( pi_b_d @ Omega
          @ ^ [Uu: b] : Omega2 ) )
     => ( ( collect_set_b
          @ ^ [Uu: set_b] :
            ? [A3: set_d] :
              ( ( Uu
                = ( inf_inf_set_b @ ( vimage_b_d @ X4 @ A3 ) @ Omega ) )
              & ( member_set_d @ A3 @ ( sigma_sigma_sets_d @ Omega2 @ M2 ) ) ) )
        = ( sigma_sigma_sets_b @ Omega
          @ ( collect_set_b
            @ ^ [Uu: set_b] :
              ? [A3: set_d] :
                ( ( Uu
                  = ( inf_inf_set_b @ ( vimage_b_d @ X4 @ A3 ) @ Omega ) )
                & ( member_set_d @ A3 @ M2 ) ) ) ) ) ) ).

% sigma_sets_vimage_commute
thf(fact_797_sigma__sets__vimage__commute,axiom,
    ! [X4: b > b,Omega: set_b,Omega2: set_b,M2: set_set_b] :
      ( ( member_b_b @ X4
        @ ( pi_b_b @ Omega
          @ ^ [Uu: b] : Omega2 ) )
     => ( ( collect_set_b
          @ ^ [Uu: set_b] :
            ? [A3: set_b] :
              ( ( Uu
                = ( inf_inf_set_b @ ( vimage_b_b @ X4 @ A3 ) @ Omega ) )
              & ( member_set_b @ A3 @ ( sigma_sigma_sets_b @ Omega2 @ M2 ) ) ) )
        = ( sigma_sigma_sets_b @ Omega
          @ ( collect_set_b
            @ ^ [Uu: set_b] :
              ? [A3: set_b] :
                ( ( Uu
                  = ( inf_inf_set_b @ ( vimage_b_b @ X4 @ A3 ) @ Omega ) )
                & ( member_set_b @ A3 @ M2 ) ) ) ) ) ) ).

% sigma_sets_vimage_commute
thf(fact_798_sigma__sets__vimage__commute,axiom,
    ! [X4: d > b,Omega: set_d,Omega2: set_b,M2: set_set_b] :
      ( ( member_d_b @ X4
        @ ( pi_d_b @ Omega
          @ ^ [Uu: d] : Omega2 ) )
     => ( ( collect_set_d
          @ ^ [Uu: set_d] :
            ? [A3: set_b] :
              ( ( Uu
                = ( inf_inf_set_d @ ( vimage_d_b @ X4 @ A3 ) @ Omega ) )
              & ( member_set_b @ A3 @ ( sigma_sigma_sets_b @ Omega2 @ M2 ) ) ) )
        = ( sigma_sigma_sets_d @ Omega
          @ ( collect_set_d
            @ ^ [Uu: set_d] :
              ? [A3: set_b] :
                ( ( Uu
                  = ( inf_inf_set_d @ ( vimage_d_b @ X4 @ A3 ) @ Omega ) )
                & ( member_set_b @ A3 @ M2 ) ) ) ) ) ) ).

% sigma_sets_vimage_commute
thf(fact_799_sigma__sets__vimage__commute,axiom,
    ! [X4: d > d,Omega: set_d,Omega2: set_d,M2: set_set_d] :
      ( ( member_d_d @ X4
        @ ( pi_d_d @ Omega
          @ ^ [Uu: d] : Omega2 ) )
     => ( ( collect_set_d
          @ ^ [Uu: set_d] :
            ? [A3: set_d] :
              ( ( Uu
                = ( inf_inf_set_d @ ( vimage_d_d @ X4 @ A3 ) @ Omega ) )
              & ( member_set_d @ A3 @ ( sigma_sigma_sets_d @ Omega2 @ M2 ) ) ) )
        = ( sigma_sigma_sets_d @ Omega
          @ ( collect_set_d
            @ ^ [Uu: set_d] :
              ? [A3: set_d] :
                ( ( Uu
                  = ( inf_inf_set_d @ ( vimage_d_d @ X4 @ A3 ) @ Omega ) )
                & ( member_set_d @ A3 @ M2 ) ) ) ) ) ) ).

% sigma_sets_vimage_commute
thf(fact_800_sigma__sets__vimage__commute,axiom,
    ! [X4: set_b > d,Omega: set_set_b,Omega2: set_d,M2: set_set_d] :
      ( ( member_set_b_d @ X4
        @ ( pi_set_b_d @ Omega
          @ ^ [Uu: set_b] : Omega2 ) )
     => ( ( collect_set_set_b
          @ ^ [Uu: set_set_b] :
            ? [A3: set_d] :
              ( ( Uu
                = ( inf_inf_set_set_b @ ( vimage_set_b_d @ X4 @ A3 ) @ Omega ) )
              & ( member_set_d @ A3 @ ( sigma_sigma_sets_d @ Omega2 @ M2 ) ) ) )
        = ( sigma_2987359972167793591_set_b @ Omega
          @ ( collect_set_set_b
            @ ^ [Uu: set_set_b] :
              ? [A3: set_d] :
                ( ( Uu
                  = ( inf_inf_set_set_b @ ( vimage_set_b_d @ X4 @ A3 ) @ Omega ) )
                & ( member_set_d @ A3 @ M2 ) ) ) ) ) ) ).

% sigma_sets_vimage_commute
thf(fact_801_sigma__sets__vimage__commute,axiom,
    ! [X4: set_b > b,Omega: set_set_b,Omega2: set_b,M2: set_set_b] :
      ( ( member_set_b_b @ X4
        @ ( pi_set_b_b @ Omega
          @ ^ [Uu: set_b] : Omega2 ) )
     => ( ( collect_set_set_b
          @ ^ [Uu: set_set_b] :
            ? [A3: set_b] :
              ( ( Uu
                = ( inf_inf_set_set_b @ ( vimage_set_b_b @ X4 @ A3 ) @ Omega ) )
              & ( member_set_b @ A3 @ ( sigma_sigma_sets_b @ Omega2 @ M2 ) ) ) )
        = ( sigma_2987359972167793591_set_b @ Omega
          @ ( collect_set_set_b
            @ ^ [Uu: set_set_b] :
              ? [A3: set_b] :
                ( ( Uu
                  = ( inf_inf_set_set_b @ ( vimage_set_b_b @ X4 @ A3 ) @ Omega ) )
                & ( member_set_b @ A3 @ M2 ) ) ) ) ) ) ).

% sigma_sets_vimage_commute
thf(fact_802_sets_Osigma__sets__eq,axiom,
    ! [M: sigma_measure_b] :
      ( ( sigma_sigma_sets_b @ ( sigma_space_b @ M ) @ ( sigma_sets_b @ M ) )
      = ( sigma_sets_b @ M ) ) ).

% sets.sigma_sets_eq
thf(fact_803_sets_Osigma__sets__eq,axiom,
    ! [M: sigma_measure_d] :
      ( ( sigma_sigma_sets_d @ ( sigma_space_d @ M ) @ ( sigma_sets_d @ M ) )
      = ( sigma_sets_d @ M ) ) ).

% sets.sigma_sets_eq
thf(fact_804_sets_Osigma__sets__subset_H,axiom,
    ! [A2: set_set_d,M: sigma_measure_d,Omega2: set_d] :
      ( ( ord_le3937772867789547105_set_d @ A2 @ ( sigma_sets_d @ M ) )
     => ( ( member_set_d @ Omega2 @ ( sigma_sets_d @ M ) )
       => ( ord_le3937772867789547105_set_d @ ( sigma_sigma_sets_d @ Omega2 @ A2 ) @ ( sigma_sets_d @ M ) ) ) ) ).

% sets.sigma_sets_subset'
thf(fact_805_sets_Osigma__sets__subset_H,axiom,
    ! [A2: set_set_b,M: sigma_measure_b,Omega2: set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b @ Omega2 @ ( sigma_sets_b @ M ) )
       => ( ord_le3795704787696855135_set_b @ ( sigma_sigma_sets_b @ Omega2 @ A2 ) @ ( sigma_sets_b @ M ) ) ) ) ).

% sets.sigma_sets_subset'
thf(fact_806_emeasure__mono,axiom,
    ! [A2: set_d,B2: set_d,M: sigma_measure_d] :
      ( ( ord_less_eq_set_d @ A2 @ B2 )
     => ( ( member_set_d @ B2 @ ( sigma_sets_d @ M ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_d @ M @ A2 ) @ ( sigma_emeasure_d @ M @ B2 ) ) ) ) ).

% emeasure_mono
thf(fact_807_emeasure__mono,axiom,
    ! [A2: set_set_b,B2: set_set_b,M: sigma_measure_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ B2 )
     => ( ( member_set_set_b @ B2 @ ( sigma_sets_set_b @ M ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_set_b @ M @ A2 ) @ ( sigma_emeasure_set_b @ M @ B2 ) ) ) ) ).

% emeasure_mono
thf(fact_808_emeasure__mono,axiom,
    ! [A2: set_b,B2: set_b,M: sigma_measure_b] :
      ( ( ord_less_eq_set_b @ A2 @ B2 )
     => ( ( member_set_b @ B2 @ ( sigma_sets_b @ M ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_b @ M @ A2 ) @ ( sigma_emeasure_b @ M @ B2 ) ) ) ) ).

% emeasure_mono
thf(fact_809_vimage__algebra__vimage__algebra__eq,axiom,
    ! [F: b > d,X4: set_b,Y2: set_d,G: d > b,M: sigma_measure_b] :
      ( ( member_b_d @ F
        @ ( pi_b_d @ X4
          @ ^ [Uu: b] : Y2 ) )
     => ( ( member_d_b @ G
          @ ( pi_d_b @ Y2
            @ ^ [Uu: d] : ( sigma_space_b @ M ) ) )
       => ( ( sigma_8454314777852479830ra_b_d @ X4 @ F @ ( sigma_2879403613907127762ra_d_b @ Y2 @ G @ M ) )
          = ( sigma_8454314777852479828ra_b_b @ X4
            @ ^ [X2: b] : ( G @ ( F @ X2 ) )
            @ M ) ) ) ) ).

% vimage_algebra_vimage_algebra_eq
thf(fact_810_vimage__algebra__vimage__algebra__eq,axiom,
    ! [F: b > b,X4: set_b,Y2: set_b,G: b > b,M: sigma_measure_b] :
      ( ( member_b_b @ F
        @ ( pi_b_b @ X4
          @ ^ [Uu: b] : Y2 ) )
     => ( ( member_b_b @ G
          @ ( pi_b_b @ Y2
            @ ^ [Uu: b] : ( sigma_space_b @ M ) ) )
       => ( ( sigma_8454314777852479828ra_b_b @ X4 @ F @ ( sigma_8454314777852479828ra_b_b @ Y2 @ G @ M ) )
          = ( sigma_8454314777852479828ra_b_b @ X4
            @ ^ [X2: b] : ( G @ ( F @ X2 ) )
            @ M ) ) ) ) ).

% vimage_algebra_vimage_algebra_eq
thf(fact_811_vimage__algebra__vimage__algebra__eq,axiom,
    ! [F: d > b,X4: set_d,Y2: set_b,G: b > b,M: sigma_measure_b] :
      ( ( member_d_b @ F
        @ ( pi_d_b @ X4
          @ ^ [Uu: d] : Y2 ) )
     => ( ( member_b_b @ G
          @ ( pi_b_b @ Y2
            @ ^ [Uu: b] : ( sigma_space_b @ M ) ) )
       => ( ( sigma_2879403613907127762ra_d_b @ X4 @ F @ ( sigma_8454314777852479828ra_b_b @ Y2 @ G @ M ) )
          = ( sigma_2879403613907127762ra_d_b @ X4
            @ ^ [X2: d] : ( G @ ( F @ X2 ) )
            @ M ) ) ) ) ).

% vimage_algebra_vimage_algebra_eq
thf(fact_812_vimage__algebra__vimage__algebra__eq,axiom,
    ! [F: d > d,X4: set_d,Y2: set_d,G: d > b,M: sigma_measure_b] :
      ( ( member_d_d @ F
        @ ( pi_d_d @ X4
          @ ^ [Uu: d] : Y2 ) )
     => ( ( member_d_b @ G
          @ ( pi_d_b @ Y2
            @ ^ [Uu: d] : ( sigma_space_b @ M ) ) )
       => ( ( sigma_2879403613907127764ra_d_d @ X4 @ F @ ( sigma_2879403613907127762ra_d_b @ Y2 @ G @ M ) )
          = ( sigma_2879403613907127762ra_d_b @ X4
            @ ^ [X2: d] : ( G @ ( F @ X2 ) )
            @ M ) ) ) ) ).

% vimage_algebra_vimage_algebra_eq
thf(fact_813_vimage__algebra__vimage__algebra__eq,axiom,
    ! [F: b > d,X4: set_b,Y2: set_d,G: d > d,M: sigma_measure_d] :
      ( ( member_b_d @ F
        @ ( pi_b_d @ X4
          @ ^ [Uu: b] : Y2 ) )
     => ( ( member_d_d @ G
          @ ( pi_d_d @ Y2
            @ ^ [Uu: d] : ( sigma_space_d @ M ) ) )
       => ( ( sigma_8454314777852479830ra_b_d @ X4 @ F @ ( sigma_2879403613907127764ra_d_d @ Y2 @ G @ M ) )
          = ( sigma_8454314777852479830ra_b_d @ X4
            @ ^ [X2: b] : ( G @ ( F @ X2 ) )
            @ M ) ) ) ) ).

% vimage_algebra_vimage_algebra_eq
thf(fact_814_vimage__algebra__vimage__algebra__eq,axiom,
    ! [F: b > b,X4: set_b,Y2: set_b,G: b > d,M: sigma_measure_d] :
      ( ( member_b_b @ F
        @ ( pi_b_b @ X4
          @ ^ [Uu: b] : Y2 ) )
     => ( ( member_b_d @ G
          @ ( pi_b_d @ Y2
            @ ^ [Uu: b] : ( sigma_space_d @ M ) ) )
       => ( ( sigma_8454314777852479828ra_b_b @ X4 @ F @ ( sigma_8454314777852479830ra_b_d @ Y2 @ G @ M ) )
          = ( sigma_8454314777852479830ra_b_d @ X4
            @ ^ [X2: b] : ( G @ ( F @ X2 ) )
            @ M ) ) ) ) ).

% vimage_algebra_vimage_algebra_eq
thf(fact_815_vimage__algebra__vimage__algebra__eq,axiom,
    ! [F: d > b,X4: set_d,Y2: set_b,G: b > d,M: sigma_measure_d] :
      ( ( member_d_b @ F
        @ ( pi_d_b @ X4
          @ ^ [Uu: d] : Y2 ) )
     => ( ( member_b_d @ G
          @ ( pi_b_d @ Y2
            @ ^ [Uu: b] : ( sigma_space_d @ M ) ) )
       => ( ( sigma_2879403613907127762ra_d_b @ X4 @ F @ ( sigma_8454314777852479830ra_b_d @ Y2 @ G @ M ) )
          = ( sigma_2879403613907127764ra_d_d @ X4
            @ ^ [X2: d] : ( G @ ( F @ X2 ) )
            @ M ) ) ) ) ).

% vimage_algebra_vimage_algebra_eq
thf(fact_816_vimage__algebra__vimage__algebra__eq,axiom,
    ! [F: d > d,X4: set_d,Y2: set_d,G: d > d,M: sigma_measure_d] :
      ( ( member_d_d @ F
        @ ( pi_d_d @ X4
          @ ^ [Uu: d] : Y2 ) )
     => ( ( member_d_d @ G
          @ ( pi_d_d @ Y2
            @ ^ [Uu: d] : ( sigma_space_d @ M ) ) )
       => ( ( sigma_2879403613907127764ra_d_d @ X4 @ F @ ( sigma_2879403613907127764ra_d_d @ Y2 @ G @ M ) )
          = ( sigma_2879403613907127764ra_d_d @ X4
            @ ^ [X2: d] : ( G @ ( F @ X2 ) )
            @ M ) ) ) ) ).

% vimage_algebra_vimage_algebra_eq
thf(fact_817_le__measure,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d] :
      ( ( ( sigma_sets_d @ M )
        = ( sigma_sets_d @ N ) )
     => ( ( ord_le254669808495466590sure_d @ M @ N )
        = ( ! [X2: set_d] :
              ( ( member_set_d @ X2 @ ( sigma_sets_d @ M ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_d @ M @ X2 ) @ ( sigma_emeasure_d @ N @ X2 ) ) ) ) ) ) ).

% le_measure
thf(fact_818_le__measure,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b] :
      ( ( ( sigma_sets_b @ M )
        = ( sigma_sets_b @ N ) )
     => ( ( ord_le254669799889008988sure_b @ M @ N )
        = ( ! [X2: set_b] :
              ( ( member_set_b @ X2 @ ( sigma_sets_b @ M ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_b @ M @ X2 ) @ ( sigma_emeasure_b @ N @ X2 ) ) ) ) ) ) ).

% le_measure
thf(fact_819_le__measureD3,axiom,
    ! [A: sigma_measure_d,B: sigma_measure_d,X4: set_d] :
      ( ( ord_le254669808495466590sure_d @ A @ B )
     => ( ( ( sigma_sets_d @ A )
          = ( sigma_sets_d @ B ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_d @ A @ X4 ) @ ( sigma_emeasure_d @ B @ X4 ) ) ) ) ).

% le_measureD3
thf(fact_820_le__measureD3,axiom,
    ! [A: sigma_measure_b,B: sigma_measure_b,X4: set_b] :
      ( ( ord_le254669799889008988sure_b @ A @ B )
     => ( ( ( sigma_sets_b @ A )
          = ( sigma_sets_b @ B ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_b @ A @ X4 ) @ ( sigma_emeasure_b @ B @ X4 ) ) ) ) ).

% le_measureD3
thf(fact_821_sets_Osigma__sets__subset,axiom,
    ! [A2: set_set_d,M: sigma_measure_d] :
      ( ( ord_le3937772867789547105_set_d @ A2 @ ( sigma_sets_d @ M ) )
     => ( ord_le3937772867789547105_set_d @ ( sigma_sigma_sets_d @ ( sigma_space_d @ M ) @ A2 ) @ ( sigma_sets_d @ M ) ) ) ).

% sets.sigma_sets_subset
thf(fact_822_sets_Osigma__sets__subset,axiom,
    ! [A2: set_set_b,M: sigma_measure_b] :
      ( ( ord_le3795704787696855135_set_b @ A2 @ ( sigma_sets_b @ M ) )
     => ( ord_le3795704787696855135_set_b @ ( sigma_sigma_sets_b @ ( sigma_space_b @ M ) @ A2 ) @ ( sigma_sets_b @ M ) ) ) ).

% sets.sigma_sets_subset
thf(fact_823_sigma__sets__le__sets__iff,axiom,
    ! [X: sigma_measure_d,A7: set_set_d] :
      ( ( ord_le3937772867789547105_set_d @ ( sigma_sigma_sets_d @ ( sigma_space_d @ X ) @ A7 ) @ ( sigma_sets_d @ X ) )
      = ( ord_le3937772867789547105_set_d @ A7 @ ( sigma_sets_d @ X ) ) ) ).

% sigma_sets_le_sets_iff
thf(fact_824_sigma__sets__le__sets__iff,axiom,
    ! [X: sigma_measure_b,A7: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ ( sigma_sigma_sets_b @ ( sigma_space_b @ X ) @ A7 ) @ ( sigma_sets_b @ X ) )
      = ( ord_le3795704787696855135_set_b @ A7 @ ( sigma_sets_b @ X ) ) ) ).

% sigma_sets_le_sets_iff
thf(fact_825_measurable__vimage__algebra1,axiom,
    ! [F: b > d,X4: set_b,M: sigma_measure_d] :
      ( ( member_b_d @ F
        @ ( pi_b_d @ X4
          @ ^ [Uu: b] : ( sigma_space_d @ M ) ) )
     => ( member_b_d @ F @ ( sigma_measurable_b_d @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ M ) @ M ) ) ) ).

% measurable_vimage_algebra1
thf(fact_826_measurable__vimage__algebra1,axiom,
    ! [F: b > b,X4: set_b,M: sigma_measure_b] :
      ( ( member_b_b @ F
        @ ( pi_b_b @ X4
          @ ^ [Uu: b] : ( sigma_space_b @ M ) ) )
     => ( member_b_b @ F @ ( sigma_measurable_b_b @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ M ) @ M ) ) ) ).

% measurable_vimage_algebra1
thf(fact_827_measurable__vimage__algebra1,axiom,
    ! [F: d > b,X4: set_d,M: sigma_measure_b] :
      ( ( member_d_b @ F
        @ ( pi_d_b @ X4
          @ ^ [Uu: d] : ( sigma_space_b @ M ) ) )
     => ( member_d_b @ F @ ( sigma_measurable_d_b @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ M ) @ M ) ) ) ).

% measurable_vimage_algebra1
thf(fact_828_measurable__vimage__algebra1,axiom,
    ! [F: d > d,X4: set_d,M: sigma_measure_d] :
      ( ( member_d_d @ F
        @ ( pi_d_d @ X4
          @ ^ [Uu: d] : ( sigma_space_d @ M ) ) )
     => ( member_d_d @ F @ ( sigma_measurable_d_d @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ M ) @ M ) ) ) ).

% measurable_vimage_algebra1
thf(fact_829_measurable__vimage__algebra2,axiom,
    ! [G: b > d,N: sigma_measure_b,X4: set_d,F: d > d,M: sigma_measure_d] :
      ( ( member_b_d @ G
        @ ( pi_b_d @ ( sigma_space_b @ N )
          @ ^ [Uu: b] : X4 ) )
     => ( ( member_b_d
          @ ^ [X2: b] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_b_d @ N @ M ) )
       => ( member_b_d @ G @ ( sigma_measurable_b_d @ N @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ M ) ) ) ) ) ).

% measurable_vimage_algebra2
thf(fact_830_measurable__vimage__algebra2,axiom,
    ! [G: b > b,N: sigma_measure_b,X4: set_b,F: b > d,M: sigma_measure_d] :
      ( ( member_b_b @ G
        @ ( pi_b_b @ ( sigma_space_b @ N )
          @ ^ [Uu: b] : X4 ) )
     => ( ( member_b_d
          @ ^ [X2: b] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_b_d @ N @ M ) )
       => ( member_b_b @ G @ ( sigma_measurable_b_b @ N @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ M ) ) ) ) ) ).

% measurable_vimage_algebra2
thf(fact_831_measurable__vimage__algebra2,axiom,
    ! [G: b > d,N: sigma_measure_b,X4: set_d,F: d > b,M: sigma_measure_b] :
      ( ( member_b_d @ G
        @ ( pi_b_d @ ( sigma_space_b @ N )
          @ ^ [Uu: b] : X4 ) )
     => ( ( member_b_b
          @ ^ [X2: b] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_b_b @ N @ M ) )
       => ( member_b_d @ G @ ( sigma_measurable_b_d @ N @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ M ) ) ) ) ) ).

% measurable_vimage_algebra2
thf(fact_832_measurable__vimage__algebra2,axiom,
    ! [G: b > b,N: sigma_measure_b,X4: set_b,F: b > b,M: sigma_measure_b] :
      ( ( member_b_b @ G
        @ ( pi_b_b @ ( sigma_space_b @ N )
          @ ^ [Uu: b] : X4 ) )
     => ( ( member_b_b
          @ ^ [X2: b] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_b_b @ N @ M ) )
       => ( member_b_b @ G @ ( sigma_measurable_b_b @ N @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ M ) ) ) ) ) ).

% measurable_vimage_algebra2
thf(fact_833_measurable__vimage__algebra2,axiom,
    ! [G: d > b,N: sigma_measure_d,X4: set_b,F: b > b,M: sigma_measure_b] :
      ( ( member_d_b @ G
        @ ( pi_d_b @ ( sigma_space_d @ N )
          @ ^ [Uu: d] : X4 ) )
     => ( ( member_d_b
          @ ^ [X2: d] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_d_b @ N @ M ) )
       => ( member_d_b @ G @ ( sigma_measurable_d_b @ N @ ( sigma_8454314777852479828ra_b_b @ X4 @ F @ M ) ) ) ) ) ).

% measurable_vimage_algebra2
thf(fact_834_measurable__vimage__algebra2,axiom,
    ! [G: d > d,N: sigma_measure_d,X4: set_d,F: d > b,M: sigma_measure_b] :
      ( ( member_d_d @ G
        @ ( pi_d_d @ ( sigma_space_d @ N )
          @ ^ [Uu: d] : X4 ) )
     => ( ( member_d_b
          @ ^ [X2: d] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_d_b @ N @ M ) )
       => ( member_d_d @ G @ ( sigma_measurable_d_d @ N @ ( sigma_2879403613907127762ra_d_b @ X4 @ F @ M ) ) ) ) ) ).

% measurable_vimage_algebra2
thf(fact_835_measurable__vimage__algebra2,axiom,
    ! [G: d > b,N: sigma_measure_d,X4: set_b,F: b > d,M: sigma_measure_d] :
      ( ( member_d_b @ G
        @ ( pi_d_b @ ( sigma_space_d @ N )
          @ ^ [Uu: d] : X4 ) )
     => ( ( member_d_d
          @ ^ [X2: d] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_d_d @ N @ M ) )
       => ( member_d_b @ G @ ( sigma_measurable_d_b @ N @ ( sigma_8454314777852479830ra_b_d @ X4 @ F @ M ) ) ) ) ) ).

% measurable_vimage_algebra2
thf(fact_836_measurable__vimage__algebra2,axiom,
    ! [G: d > d,N: sigma_measure_d,X4: set_d,F: d > d,M: sigma_measure_d] :
      ( ( member_d_d @ G
        @ ( pi_d_d @ ( sigma_space_d @ N )
          @ ^ [Uu: d] : X4 ) )
     => ( ( member_d_d
          @ ^ [X2: d] : ( F @ ( G @ X2 ) )
          @ ( sigma_measurable_d_d @ N @ M ) )
       => ( member_d_d @ G @ ( sigma_measurable_d_d @ N @ ( sigma_2879403613907127764ra_d_d @ X4 @ F @ M ) ) ) ) ) ).

% measurable_vimage_algebra2
thf(fact_837_boolean__algebra__cancel_Oinf2,axiom,
    ! [B: set_b,K: set_b,B2: set_b,A2: set_b] :
      ( ( B
        = ( inf_inf_set_b @ K @ B2 ) )
     => ( ( inf_inf_set_b @ A2 @ B )
        = ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_838_boolean__algebra__cancel_Oinf2,axiom,
    ! [B: set_d,K: set_d,B2: set_d,A2: set_d] :
      ( ( B
        = ( inf_inf_set_d @ K @ B2 ) )
     => ( ( inf_inf_set_d @ A2 @ B )
        = ( inf_inf_set_d @ K @ ( inf_inf_set_d @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_839_boolean__algebra__cancel_Oinf2,axiom,
    ! [B: set_set_b,K: set_set_b,B2: set_set_b,A2: set_set_b] :
      ( ( B
        = ( inf_inf_set_set_b @ K @ B2 ) )
     => ( ( inf_inf_set_set_b @ A2 @ B )
        = ( inf_inf_set_set_b @ K @ ( inf_inf_set_set_b @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_840_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: set_b,K: set_b,A2: set_b,B2: set_b] :
      ( ( A
        = ( inf_inf_set_b @ K @ A2 ) )
     => ( ( inf_inf_set_b @ A @ B2 )
        = ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_841_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: set_d,K: set_d,A2: set_d,B2: set_d] :
      ( ( A
        = ( inf_inf_set_d @ K @ A2 ) )
     => ( ( inf_inf_set_d @ A @ B2 )
        = ( inf_inf_set_d @ K @ ( inf_inf_set_d @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_842_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: set_set_b,K: set_set_b,A2: set_set_b,B2: set_set_b] :
      ( ( A
        = ( inf_inf_set_set_b @ K @ A2 ) )
     => ( ( inf_inf_set_set_b @ A @ B2 )
        = ( inf_inf_set_set_b @ K @ ( inf_inf_set_set_b @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_843_less__eq__measure_Ointros_I3_J,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b] :
      ( ( ( sigma_space_b @ M )
        = ( sigma_space_b @ N ) )
     => ( ( ( sigma_sets_b @ M )
          = ( sigma_sets_b @ N ) )
       => ( ( ord_le6365330890766322088nnreal @ ( sigma_emeasure_b @ M ) @ ( sigma_emeasure_b @ N ) )
         => ( ord_le254669799889008988sure_b @ M @ N ) ) ) ) ).

% less_eq_measure.intros(3)
thf(fact_844_less__eq__measure_Ointros_I3_J,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d] :
      ( ( ( sigma_space_d @ M )
        = ( sigma_space_d @ N ) )
     => ( ( ( sigma_sets_d @ M )
          = ( sigma_sets_d @ N ) )
       => ( ( ord_le5694847263963582634nnreal @ ( sigma_emeasure_d @ M ) @ ( sigma_emeasure_d @ N ) )
         => ( ord_le254669808495466590sure_d @ M @ N ) ) ) ) ).

% less_eq_measure.intros(3)
thf(fact_845_Pi__mono,axiom,
    ! [A: set_c,B: c > set_b,C: c > set_b] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( ord_less_eq_set_b @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_less_eq_set_c_b @ ( pi_c_b @ A @ B ) @ ( pi_c_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_846_Pi__mono,axiom,
    ! [A: set_c,B: c > set_set_b,C: c > set_set_b] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( ord_le3795704787696855135_set_b @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le213790835969874778_set_b @ ( pi_c_set_b @ A @ B ) @ ( pi_c_set_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_847_Pi__mono,axiom,
    ! [A: set_set_d,B: set_d > set_b,C: set_d > set_b] :
      ( ! [X3: set_d] :
          ( ( member_set_d @ X3 @ A )
         => ( ord_less_eq_set_b @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le3805021600748269913et_d_b @ ( pi_set_d_b @ A @ B ) @ ( pi_set_d_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_848_Pi__mono,axiom,
    ! [A: set_set_b,B: set_b > set_b,C: set_b > set_b] :
      ( ! [X3: set_b] :
          ( ( member_set_b @ X3 @ A )
         => ( ord_less_eq_set_b @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le7727052776605854427et_b_b @ ( pi_set_b_b @ A @ B ) @ ( pi_set_b_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_849_Pi__mono,axiom,
    ! [A: set_set_d,B: set_d > set_set_b,C: set_d > set_set_b] :
      ( ! [X3: set_d] :
          ( ( member_set_d @ X3 @ A )
         => ( ord_le3795704787696855135_set_b @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le84730631212153401_set_b @ ( pi_set_d_set_b @ A @ B ) @ ( pi_set_d_set_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_850_Pi__mono,axiom,
    ! [A: set_set_b,B: set_b > set_set_b,C: set_b > set_set_b] :
      ( ! [X3: set_b] :
          ( ( member_set_b @ X3 @ A )
         => ( ord_le3795704787696855135_set_b @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le1406319396829752251_set_b @ ( pi_set_b_set_b @ A @ B ) @ ( pi_set_b_set_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_851_Pi__mono,axiom,
    ! [A: set_b_d,B: ( b > d ) > set_b,C: ( b > d ) > set_b] :
      ( ! [X3: b > d] :
          ( ( member_b_d @ X3 @ A )
         => ( ord_less_eq_set_b @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le3202560045296529359_b_d_b @ ( pi_b_d_b @ A @ B ) @ ( pi_b_d_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_852_Pi__mono,axiom,
    ! [A: set_b_b,B: ( b > b ) > set_b,C: ( b > b ) > set_b] :
      ( ! [X3: b > b] :
          ( ( member_b_b @ X3 @ A )
         => ( ord_less_eq_set_b @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le7124591221154113873_b_b_b @ ( pi_b_b_b @ A @ B ) @ ( pi_b_b_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_853_Pi__mono,axiom,
    ! [A: set_d_b,B: ( d > b ) > set_b,C: ( d > b ) > set_b] :
      ( ! [X3: d > b] :
          ( ( member_d_b @ X3 @ A )
         => ( ord_less_eq_set_b @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le1567182605434068819_d_b_b @ ( pi_d_b_b @ A @ B ) @ ( pi_d_b_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_854_Pi__mono,axiom,
    ! [A: set_d_d,B: ( d > d ) > set_b,C: ( d > d ) > set_b] :
      ( ! [X3: d > d] :
          ( ( member_d_d @ X3 @ A )
         => ( ord_less_eq_set_b @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le6868523466431260113_d_d_b @ ( pi_d_d_b @ A @ B ) @ ( pi_d_d_b @ A @ C ) ) ) ).

% Pi_mono
thf(fact_855_Pi__I,axiom,
    ! [A: set_b,F: b > d,B: b > set_d] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_d @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_b_d @ F @ ( pi_b_d @ A @ B ) ) ) ).

% Pi_I
thf(fact_856_Pi__I,axiom,
    ! [A: set_b,F: b > b,B: b > set_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_b_b @ F @ ( pi_b_b @ A @ B ) ) ) ).

% Pi_I
thf(fact_857_Pi__I,axiom,
    ! [A: set_d,F: d > b,B: d > set_b] :
      ( ! [X3: d] :
          ( ( member_d @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_d_b @ F @ ( pi_d_b @ A @ B ) ) ) ).

% Pi_I
thf(fact_858_Pi__I,axiom,
    ! [A: set_d,F: d > d,B: d > set_d] :
      ( ! [X3: d] :
          ( ( member_d @ X3 @ A )
         => ( member_d @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_d_d @ F @ ( pi_d_d @ A @ B ) ) ) ).

% Pi_I
thf(fact_859_Pi__I,axiom,
    ! [A: set_c,F: c > c,B: c > set_c] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_c @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_c_c @ F @ ( pi_c_c @ A @ B ) ) ) ).

% Pi_I
thf(fact_860_Pi__I,axiom,
    ! [A: set_set_d,F: set_d > c,B: set_d > set_c] :
      ( ! [X3: set_d] :
          ( ( member_set_d @ X3 @ A )
         => ( member_c @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_set_d_c @ F @ ( pi_set_d_c @ A @ B ) ) ) ).

% Pi_I
thf(fact_861_Pi__I,axiom,
    ! [A: set_c,F: c > set_d,B: c > set_set_d] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_set_d @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_c_set_d @ F @ ( pi_c_set_d @ A @ B ) ) ) ).

% Pi_I
thf(fact_862_Pi__I,axiom,
    ! [A: set_c,F: c > set_b,B: c > set_set_b] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_set_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_c_set_b @ F @ ( pi_c_set_b @ A @ B ) ) ) ).

% Pi_I
thf(fact_863_Pi__I,axiom,
    ! [A: set_set_b,F: set_b > c,B: set_b > set_c] :
      ( ! [X3: set_b] :
          ( ( member_set_b @ X3 @ A )
         => ( member_c @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_set_b_c @ F @ ( pi_set_b_c @ A @ B ) ) ) ).

% Pi_I
thf(fact_864_Pi__I,axiom,
    ! [A: set_set_d,F: set_d > set_d,B: set_d > set_set_d] :
      ( ! [X3: set_d] :
          ( ( member_set_d @ X3 @ A )
         => ( member_set_d @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_set_d_set_d @ F @ ( pi_set_d_set_d @ A @ B ) ) ) ).

% Pi_I
thf(fact_865_measurable__sigma__sets,axiom,
    ! [N: sigma_measure_d,Omega: set_d,A: set_set_d,F: set_b > d,M: sigma_measure_set_b] :
      ( ( ( sigma_sets_d @ N )
        = ( sigma_sigma_sets_d @ Omega @ A ) )
     => ( ( ord_le3937772867789547105_set_d @ A @ ( pow_d @ Omega ) )
       => ( ( member_set_b_d @ F
            @ ( pi_set_b_d @ ( sigma_space_set_b @ M )
              @ ^ [Uu: set_b] : Omega ) )
         => ( ! [Y5: set_d] :
                ( ( member_set_d @ Y5 @ A )
               => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ Y5 ) @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) ) )
           => ( member_set_b_d @ F @ ( sigma_3265996224607154482et_b_d @ M @ N ) ) ) ) ) ) ).

% measurable_sigma_sets
thf(fact_866_measurable__sigma__sets,axiom,
    ! [N: sigma_measure_d,Omega: set_d,A: set_set_d,F: b > d,M: sigma_measure_b] :
      ( ( ( sigma_sets_d @ N )
        = ( sigma_sigma_sets_d @ Omega @ A ) )
     => ( ( ord_le3937772867789547105_set_d @ A @ ( pow_d @ Omega ) )
       => ( ( member_b_d @ F
            @ ( pi_b_d @ ( sigma_space_b @ M )
              @ ^ [Uu: b] : Omega ) )
         => ( ! [Y5: set_d] :
                ( ( member_set_d @ Y5 @ A )
               => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_d @ F @ Y5 ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) )
           => ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) ) ) ) ) ) ).

% measurable_sigma_sets
thf(fact_867_measurable__sigma__sets,axiom,
    ! [N: sigma_measure_d,Omega: set_d,A: set_set_d,F: d > d,M: sigma_measure_d] :
      ( ( ( sigma_sets_d @ N )
        = ( sigma_sigma_sets_d @ Omega @ A ) )
     => ( ( ord_le3937772867789547105_set_d @ A @ ( pow_d @ Omega ) )
       => ( ( member_d_d @ F
            @ ( pi_d_d @ ( sigma_space_d @ M )
              @ ^ [Uu: d] : Omega ) )
         => ( ! [Y5: set_d] :
                ( ( member_set_d @ Y5 @ A )
               => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_d @ F @ Y5 ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) )
           => ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) ) ) ) ) ) ).

% measurable_sigma_sets
thf(fact_868_measurable__sigma__sets,axiom,
    ! [N: sigma_measure_b,Omega: set_b,A: set_set_b,F: set_b > b,M: sigma_measure_set_b] :
      ( ( ( sigma_sets_b @ N )
        = ( sigma_sigma_sets_b @ Omega @ A ) )
     => ( ( ord_le3795704787696855135_set_b @ A @ ( pow_b @ Omega ) )
       => ( ( member_set_b_b @ F
            @ ( pi_set_b_b @ ( sigma_space_set_b @ M )
              @ ^ [Uu: set_b] : Omega ) )
         => ( ! [Y5: set_b] :
                ( ( member_set_b @ Y5 @ A )
               => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ Y5 ) @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) ) )
           => ( member_set_b_b @ F @ ( sigma_3265996224607154480et_b_b @ M @ N ) ) ) ) ) ) ).

% measurable_sigma_sets
thf(fact_869_measurable__sigma__sets,axiom,
    ! [N: sigma_measure_b,Omega: set_b,A: set_set_b,F: b > b,M: sigma_measure_b] :
      ( ( ( sigma_sets_b @ N )
        = ( sigma_sigma_sets_b @ Omega @ A ) )
     => ( ( ord_le3795704787696855135_set_b @ A @ ( pow_b @ Omega ) )
       => ( ( member_b_b @ F
            @ ( pi_b_b @ ( sigma_space_b @ M )
              @ ^ [Uu: b] : Omega ) )
         => ( ! [Y5: set_b] :
                ( ( member_set_b @ Y5 @ A )
               => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_b @ F @ Y5 ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) )
           => ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) ) ) ) ) ) ).

% measurable_sigma_sets
thf(fact_870_measurable__sigma__sets,axiom,
    ! [N: sigma_measure_b,Omega: set_b,A: set_set_b,F: d > b,M: sigma_measure_d] :
      ( ( ( sigma_sets_b @ N )
        = ( sigma_sigma_sets_b @ Omega @ A ) )
     => ( ( ord_le3795704787696855135_set_b @ A @ ( pow_b @ Omega ) )
       => ( ( member_d_b @ F
            @ ( pi_d_b @ ( sigma_space_d @ M )
              @ ^ [Uu: d] : Omega ) )
         => ( ! [Y5: set_b] :
                ( ( member_set_b @ Y5 @ A )
               => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_b @ F @ Y5 ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) )
           => ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) ) ) ) ) ) ).

% measurable_sigma_sets
thf(fact_871_sets_Osigma__property__disjoint__lemma,axiom,
    ! [M: sigma_measure_d,C: set_set_d] :
      ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M ) @ C )
     => ( ( sigma_closed_cdi_d @ ( sigma_space_d @ M ) @ C )
       => ( ord_le3937772867789547105_set_d @ ( sigma_sigma_sets_d @ ( sigma_space_d @ M ) @ ( sigma_sets_d @ M ) ) @ C ) ) ) ).

% sets.sigma_property_disjoint_lemma
thf(fact_872_sets_Osigma__property__disjoint__lemma,axiom,
    ! [M: sigma_measure_b,C: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M ) @ C )
     => ( ( sigma_closed_cdi_b @ ( sigma_space_b @ M ) @ C )
       => ( ord_le3795704787696855135_set_b @ ( sigma_sigma_sets_b @ ( sigma_space_b @ M ) @ ( sigma_sets_b @ M ) ) @ C ) ) ) ).

% sets.sigma_property_disjoint_lemma
thf(fact_873_measurable__def,axiom,
    ( sigma_3265996224607154480et_b_b
    = ( ^ [A3: sigma_measure_set_b,B3: sigma_measure_b] :
          ( collect_set_b_b
          @ ^ [F2: set_b > b] :
              ( ( member_set_b_b @ F2
                @ ( pi_set_b_b @ ( sigma_space_set_b @ A3 )
                  @ ^ [Uu: set_b] : ( sigma_space_b @ B3 ) ) )
              & ! [X2: set_b] :
                  ( ( member_set_b @ X2 @ ( sigma_sets_b @ B3 ) )
                 => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F2 @ X2 ) @ ( sigma_space_set_b @ A3 ) ) @ ( sigma_sets_set_b @ A3 ) ) ) ) ) ) ) ).

% measurable_def
thf(fact_874_measurable__def,axiom,
    ( sigma_3265996224607154482et_b_d
    = ( ^ [A3: sigma_measure_set_b,B3: sigma_measure_d] :
          ( collect_set_b_d
          @ ^ [F2: set_b > d] :
              ( ( member_set_b_d @ F2
                @ ( pi_set_b_d @ ( sigma_space_set_b @ A3 )
                  @ ^ [Uu: set_b] : ( sigma_space_d @ B3 ) ) )
              & ! [X2: set_d] :
                  ( ( member_set_d @ X2 @ ( sigma_sets_d @ B3 ) )
                 => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_d @ F2 @ X2 ) @ ( sigma_space_set_b @ A3 ) ) @ ( sigma_sets_set_b @ A3 ) ) ) ) ) ) ) ).

% measurable_def
thf(fact_875_measurable__def,axiom,
    ( sigma_measurable_b_d
    = ( ^ [A3: sigma_measure_b,B3: sigma_measure_d] :
          ( collect_b_d
          @ ^ [F2: b > d] :
              ( ( member_b_d @ F2
                @ ( pi_b_d @ ( sigma_space_b @ A3 )
                  @ ^ [Uu: b] : ( sigma_space_d @ B3 ) ) )
              & ! [X2: set_d] :
                  ( ( member_set_d @ X2 @ ( sigma_sets_d @ B3 ) )
                 => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_d @ F2 @ X2 ) @ ( sigma_space_b @ A3 ) ) @ ( sigma_sets_b @ A3 ) ) ) ) ) ) ) ).

% measurable_def
thf(fact_876_measurable__def,axiom,
    ( sigma_measurable_b_b
    = ( ^ [A3: sigma_measure_b,B3: sigma_measure_b] :
          ( collect_b_b
          @ ^ [F2: b > b] :
              ( ( member_b_b @ F2
                @ ( pi_b_b @ ( sigma_space_b @ A3 )
                  @ ^ [Uu: b] : ( sigma_space_b @ B3 ) ) )
              & ! [X2: set_b] :
                  ( ( member_set_b @ X2 @ ( sigma_sets_b @ B3 ) )
                 => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_b @ F2 @ X2 ) @ ( sigma_space_b @ A3 ) ) @ ( sigma_sets_b @ A3 ) ) ) ) ) ) ) ).

% measurable_def
thf(fact_877_measurable__def,axiom,
    ( sigma_measurable_d_b
    = ( ^ [A3: sigma_measure_d,B3: sigma_measure_b] :
          ( collect_d_b
          @ ^ [F2: d > b] :
              ( ( member_d_b @ F2
                @ ( pi_d_b @ ( sigma_space_d @ A3 )
                  @ ^ [Uu: d] : ( sigma_space_b @ B3 ) ) )
              & ! [X2: set_b] :
                  ( ( member_set_b @ X2 @ ( sigma_sets_b @ B3 ) )
                 => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_b @ F2 @ X2 ) @ ( sigma_space_d @ A3 ) ) @ ( sigma_sets_d @ A3 ) ) ) ) ) ) ) ).

% measurable_def
thf(fact_878_measurable__def,axiom,
    ( sigma_measurable_d_d
    = ( ^ [A3: sigma_measure_d,B3: sigma_measure_d] :
          ( collect_d_d
          @ ^ [F2: d > d] :
              ( ( member_d_d @ F2
                @ ( pi_d_d @ ( sigma_space_d @ A3 )
                  @ ^ [Uu: d] : ( sigma_space_d @ B3 ) ) )
              & ! [X2: set_d] :
                  ( ( member_set_d @ X2 @ ( sigma_sets_d @ B3 ) )
                 => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_d @ F2 @ X2 ) @ ( sigma_space_d @ A3 ) ) @ ( sigma_sets_d @ A3 ) ) ) ) ) ) ) ).

% measurable_def
thf(fact_879_emeasure__distr,axiom,
    ! [F: set_b > d,M: sigma_measure_set_b,N: sigma_measure_d,A: set_d] :
      ( ( member_set_b_d @ F @ ( sigma_3265996224607154482et_b_d @ M @ N ) )
     => ( ( member_set_d @ A @ ( sigma_sets_d @ N ) )
       => ( ( sigma_emeasure_d @ ( measur6428830690898164186et_b_d @ M @ N @ F ) @ A )
          = ( sigma_emeasure_set_b @ M @ ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ A ) @ ( sigma_space_set_b @ M ) ) ) ) ) ) ).

% emeasure_distr
thf(fact_880_emeasure__distr,axiom,
    ! [F: set_b > b,M: sigma_measure_set_b,N: sigma_measure_b,A: set_b] :
      ( ( member_set_b_b @ F @ ( sigma_3265996224607154480et_b_b @ M @ N ) )
     => ( ( member_set_b @ A @ ( sigma_sets_b @ N ) )
       => ( ( sigma_emeasure_b @ ( measur6428830690898164184et_b_b @ M @ N @ F ) @ A )
          = ( sigma_emeasure_set_b @ M @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ A ) @ ( sigma_space_set_b @ M ) ) ) ) ) ) ).

% emeasure_distr
thf(fact_881_emeasure__distr,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d,A: set_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
     => ( ( member_set_d @ A @ ( sigma_sets_d @ N ) )
       => ( ( sigma_emeasure_d @ ( measure_distr_b_d @ M @ N @ F ) @ A )
          = ( sigma_emeasure_b @ M @ ( inf_inf_set_b @ ( vimage_b_d @ F @ A ) @ ( sigma_space_b @ M ) ) ) ) ) ) ).

% emeasure_distr
thf(fact_882_emeasure__distr,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,A: set_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( ( member_set_b @ A @ ( sigma_sets_b @ N ) )
       => ( ( sigma_emeasure_b @ ( measure_distr_b_b @ M @ N @ F ) @ A )
          = ( sigma_emeasure_b @ M @ ( inf_inf_set_b @ ( vimage_b_b @ F @ A ) @ ( sigma_space_b @ M ) ) ) ) ) ) ).

% emeasure_distr
thf(fact_883_emeasure__distr,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b,A: set_b] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
     => ( ( member_set_b @ A @ ( sigma_sets_b @ N ) )
       => ( ( sigma_emeasure_b @ ( measure_distr_d_b @ M @ N @ F ) @ A )
          = ( sigma_emeasure_d @ M @ ( inf_inf_set_d @ ( vimage_d_b @ F @ A ) @ ( sigma_space_d @ M ) ) ) ) ) ) ).

% emeasure_distr
thf(fact_884_emeasure__distr,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d,A: set_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
     => ( ( member_set_d @ A @ ( sigma_sets_d @ N ) )
       => ( ( sigma_emeasure_d @ ( measure_distr_d_d @ M @ N @ F ) @ A )
          = ( sigma_emeasure_d @ M @ ( inf_inf_set_d @ ( vimage_d_d @ F @ A ) @ ( sigma_space_d @ M ) ) ) ) ) ) ).

% emeasure_distr
thf(fact_885_PowI,axiom,
    ! [A: set_d,B: set_d] :
      ( ( ord_less_eq_set_d @ A @ B )
     => ( member_set_d @ A @ ( pow_d @ B ) ) ) ).

% PowI
thf(fact_886_PowI,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( member_set_set_b @ A @ ( pow_set_b @ B ) ) ) ).

% PowI
thf(fact_887_PowI,axiom,
    ! [A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( member_set_b @ A @ ( pow_b @ B ) ) ) ).

% PowI
thf(fact_888_Pow__iff,axiom,
    ! [A: set_d,B: set_d] :
      ( ( member_set_d @ A @ ( pow_d @ B ) )
      = ( ord_less_eq_set_d @ A @ B ) ) ).

% Pow_iff
thf(fact_889_Pow__iff,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( member_set_set_b @ A @ ( pow_set_b @ B ) )
      = ( ord_le3795704787696855135_set_b @ A @ B ) ) ).

% Pow_iff
thf(fact_890_Pow__iff,axiom,
    ! [A: set_b,B: set_b] :
      ( ( member_set_b @ A @ ( pow_b @ B ) )
      = ( ord_less_eq_set_b @ A @ B ) ) ).

% Pow_iff
thf(fact_891_Pow__Int__eq,axiom,
    ! [A: set_b,B: set_b] :
      ( ( pow_b @ ( inf_inf_set_b @ A @ B ) )
      = ( inf_inf_set_set_b @ ( pow_b @ A ) @ ( pow_b @ B ) ) ) ).

% Pow_Int_eq
thf(fact_892_Pow__Int__eq,axiom,
    ! [A: set_d,B: set_d] :
      ( ( pow_d @ ( inf_inf_set_d @ A @ B ) )
      = ( inf_inf_set_set_d @ ( pow_d @ A ) @ ( pow_d @ B ) ) ) ).

% Pow_Int_eq
thf(fact_893_Pow__Int__eq,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( pow_set_b @ ( inf_inf_set_set_b @ A @ B ) )
      = ( inf_in7907464295584518029_set_b @ ( pow_set_b @ A ) @ ( pow_set_b @ B ) ) ) ).

% Pow_Int_eq
thf(fact_894_closed__cdi__subset,axiom,
    ! [Omega: set_b,M: set_set_b] :
      ( ( sigma_closed_cdi_b @ Omega @ M )
     => ( ord_le3795704787696855135_set_b @ M @ ( pow_b @ Omega ) ) ) ).

% closed_cdi_subset
thf(fact_895_Pow__top,axiom,
    ! [A: set_d] : ( member_set_d @ A @ ( pow_d @ A ) ) ).

% Pow_top
thf(fact_896_Pow__top,axiom,
    ! [A: set_b] : ( member_set_b @ A @ ( pow_b @ A ) ) ).

% Pow_top
thf(fact_897_Ball__def,axiom,
    ( ball_set_d
    = ( ^ [A3: set_set_d,P2: set_d > $o] :
        ! [X2: set_d] :
          ( ( member_set_d @ X2 @ A3 )
         => ( P2 @ X2 ) ) ) ) ).

% Ball_def
thf(fact_898_Ball__def,axiom,
    ( ball_c
    = ( ^ [A3: set_c,P2: c > $o] :
        ! [X2: c] :
          ( ( member_c @ X2 @ A3 )
         => ( P2 @ X2 ) ) ) ) ).

% Ball_def
thf(fact_899_Ball__def,axiom,
    ( ball_b_d
    = ( ^ [A3: set_b_d,P2: ( b > d ) > $o] :
        ! [X2: b > d] :
          ( ( member_b_d @ X2 @ A3 )
         => ( P2 @ X2 ) ) ) ) ).

% Ball_def
thf(fact_900_Ball__def,axiom,
    ( ball_set_b
    = ( ^ [A3: set_set_b,P2: set_b > $o] :
        ! [X2: set_b] :
          ( ( member_set_b @ X2 @ A3 )
         => ( P2 @ X2 ) ) ) ) ).

% Ball_def
thf(fact_901_Ball__def,axiom,
    ( ball_b_b
    = ( ^ [A3: set_b_b,P2: ( b > b ) > $o] :
        ! [X2: b > b] :
          ( ( member_b_b @ X2 @ A3 )
         => ( P2 @ X2 ) ) ) ) ).

% Ball_def
thf(fact_902_Ball__def,axiom,
    ( ball_d_b
    = ( ^ [A3: set_d_b,P2: ( d > b ) > $o] :
        ! [X2: d > b] :
          ( ( member_d_b @ X2 @ A3 )
         => ( P2 @ X2 ) ) ) ) ).

% Ball_def
thf(fact_903_Ball__def,axiom,
    ( ball_d_d
    = ( ^ [A3: set_d_d,P2: ( d > d ) > $o] :
        ! [X2: d > d] :
          ( ( member_d_d @ X2 @ A3 )
         => ( P2 @ X2 ) ) ) ) ).

% Ball_def
thf(fact_904_PowD,axiom,
    ! [A: set_d,B: set_d] :
      ( ( member_set_d @ A @ ( pow_d @ B ) )
     => ( ord_less_eq_set_d @ A @ B ) ) ).

% PowD
thf(fact_905_PowD,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( member_set_set_b @ A @ ( pow_set_b @ B ) )
     => ( ord_le3795704787696855135_set_b @ A @ B ) ) ).

% PowD
thf(fact_906_PowD,axiom,
    ! [A: set_b,B: set_b] :
      ( ( member_set_b @ A @ ( pow_b @ B ) )
     => ( ord_less_eq_set_b @ A @ B ) ) ).

% PowD
thf(fact_907_distr__id2,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d] :
      ( ( ( sigma_sets_d @ M )
        = ( sigma_sets_d @ N ) )
     => ( ( measure_distr_d_d @ N @ M
          @ ^ [X2: d] : X2 )
        = N ) ) ).

% distr_id2
thf(fact_908_distr__id2,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b] :
      ( ( ( sigma_sets_b @ M )
        = ( sigma_sets_b @ N ) )
     => ( ( measure_distr_b_b @ N @ M
          @ ^ [X2: b] : X2 )
        = N ) ) ).

% distr_id2
thf(fact_909_Pow__def,axiom,
    ( pow_set_b
    = ( ^ [A3: set_set_b] :
          ( collect_set_set_b
          @ ^ [B3: set_set_b] : ( ord_le3795704787696855135_set_b @ B3 @ A3 ) ) ) ) ).

% Pow_def
thf(fact_910_Pow__def,axiom,
    ( pow_b
    = ( ^ [A3: set_b] :
          ( collect_set_b
          @ ^ [B3: set_b] : ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).

% Pow_def
thf(fact_911_distr__cong,axiom,
    ! [M: sigma_measure_c,K2: sigma_measure_c,N: sigma_measure_d,L: sigma_measure_d,F: c > d,G: c > d] :
      ( ( M = K2 )
     => ( ( ( sigma_sets_d @ N )
          = ( sigma_sets_d @ L ) )
       => ( ! [X3: c] :
              ( ( member_c @ X3 @ ( sigma_space_c @ M ) )
             => ( ( F @ X3 )
                = ( G @ X3 ) ) )
         => ( ( measure_distr_c_d @ M @ N @ F )
            = ( measure_distr_c_d @ K2 @ L @ G ) ) ) ) ) ).

% distr_cong
thf(fact_912_distr__cong,axiom,
    ! [M: sigma_measure_c,K2: sigma_measure_c,N: sigma_measure_b,L: sigma_measure_b,F: c > b,G: c > b] :
      ( ( M = K2 )
     => ( ( ( sigma_sets_b @ N )
          = ( sigma_sets_b @ L ) )
       => ( ! [X3: c] :
              ( ( member_c @ X3 @ ( sigma_space_c @ M ) )
             => ( ( F @ X3 )
                = ( G @ X3 ) ) )
         => ( ( measure_distr_c_b @ M @ N @ F )
            = ( measure_distr_c_b @ K2 @ L @ G ) ) ) ) ) ).

% distr_cong
thf(fact_913_distr__cong,axiom,
    ! [M: sigma_measure_b,K2: sigma_measure_b,N: sigma_measure_d,L: sigma_measure_d,F: b > d,G: b > d] :
      ( ( M = K2 )
     => ( ( ( sigma_sets_d @ N )
          = ( sigma_sets_d @ L ) )
       => ( ! [X3: b] :
              ( ( member_b @ X3 @ ( sigma_space_b @ M ) )
             => ( ( F @ X3 )
                = ( G @ X3 ) ) )
         => ( ( measure_distr_b_d @ M @ N @ F )
            = ( measure_distr_b_d @ K2 @ L @ G ) ) ) ) ) ).

% distr_cong
thf(fact_914_distr__cong,axiom,
    ! [M: sigma_measure_b,K2: sigma_measure_b,N: sigma_measure_b,L: sigma_measure_b,F: b > b,G: b > b] :
      ( ( M = K2 )
     => ( ( ( sigma_sets_b @ N )
          = ( sigma_sets_b @ L ) )
       => ( ! [X3: b] :
              ( ( member_b @ X3 @ ( sigma_space_b @ M ) )
             => ( ( F @ X3 )
                = ( G @ X3 ) ) )
         => ( ( measure_distr_b_b @ M @ N @ F )
            = ( measure_distr_b_b @ K2 @ L @ G ) ) ) ) ) ).

% distr_cong
thf(fact_915_distr__cong,axiom,
    ! [M: sigma_measure_d,K2: sigma_measure_d,N: sigma_measure_d,L: sigma_measure_d,F: d > d,G: d > d] :
      ( ( M = K2 )
     => ( ( ( sigma_sets_d @ N )
          = ( sigma_sets_d @ L ) )
       => ( ! [X3: d] :
              ( ( member_d @ X3 @ ( sigma_space_d @ M ) )
             => ( ( F @ X3 )
                = ( G @ X3 ) ) )
         => ( ( measure_distr_d_d @ M @ N @ F )
            = ( measure_distr_d_d @ K2 @ L @ G ) ) ) ) ) ).

% distr_cong
thf(fact_916_distr__cong,axiom,
    ! [M: sigma_measure_d,K2: sigma_measure_d,N: sigma_measure_b,L: sigma_measure_b,F: d > b,G: d > b] :
      ( ( M = K2 )
     => ( ( ( sigma_sets_b @ N )
          = ( sigma_sets_b @ L ) )
       => ( ! [X3: d] :
              ( ( member_d @ X3 @ ( sigma_space_d @ M ) )
             => ( ( F @ X3 )
                = ( G @ X3 ) ) )
         => ( ( measure_distr_d_b @ M @ N @ F )
            = ( measure_distr_d_b @ K2 @ L @ G ) ) ) ) ) ).

% distr_cong
thf(fact_917_distr__cong,axiom,
    ! [M: sigma_measure_set_d,K2: sigma_measure_set_d,N: sigma_measure_d,L: sigma_measure_d,F: set_d > d,G: set_d > d] :
      ( ( M = K2 )
     => ( ( ( sigma_sets_d @ N )
          = ( sigma_sets_d @ L ) )
       => ( ! [X3: set_d] :
              ( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
             => ( ( F @ X3 )
                = ( G @ X3 ) ) )
         => ( ( measur5157532689688190808et_d_d @ M @ N @ F )
            = ( measur5157532689688190808et_d_d @ K2 @ L @ G ) ) ) ) ) ).

% distr_cong
thf(fact_918_distr__cong,axiom,
    ! [M: sigma_measure_set_b,K2: sigma_measure_set_b,N: sigma_measure_d,L: sigma_measure_d,F: set_b > d,G: set_b > d] :
      ( ( M = K2 )
     => ( ( ( sigma_sets_d @ N )
          = ( sigma_sets_d @ L ) )
       => ( ! [X3: set_b] :
              ( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
             => ( ( F @ X3 )
                = ( G @ X3 ) ) )
         => ( ( measur6428830690898164186et_b_d @ M @ N @ F )
            = ( measur6428830690898164186et_b_d @ K2 @ L @ G ) ) ) ) ) ).

% distr_cong
thf(fact_919_distr__cong,axiom,
    ! [M: sigma_measure_set_d,K2: sigma_measure_set_d,N: sigma_measure_b,L: sigma_measure_b,F: set_d > b,G: set_d > b] :
      ( ( M = K2 )
     => ( ( ( sigma_sets_b @ N )
          = ( sigma_sets_b @ L ) )
       => ( ! [X3: set_d] :
              ( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
             => ( ( F @ X3 )
                = ( G @ X3 ) ) )
         => ( ( measur5157532689688190806et_d_b @ M @ N @ F )
            = ( measur5157532689688190806et_d_b @ K2 @ L @ G ) ) ) ) ) ).

% distr_cong
thf(fact_920_distr__cong,axiom,
    ! [M: sigma_measure_set_b,K2: sigma_measure_set_b,N: sigma_measure_b,L: sigma_measure_b,F: set_b > b,G: set_b > b] :
      ( ( M = K2 )
     => ( ( ( sigma_sets_b @ N )
          = ( sigma_sets_b @ L ) )
       => ( ! [X3: set_b] :
              ( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
             => ( ( F @ X3 )
                = ( G @ X3 ) ) )
         => ( ( measur6428830690898164184et_b_b @ M @ N @ F )
            = ( measur6428830690898164184et_b_b @ K2 @ L @ G ) ) ) ) ) ).

% distr_cong
thf(fact_921_Pow__mono,axiom,
    ! [A: set_set_b,B: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ B )
     => ( ord_le3201067847557142847_set_b @ ( pow_set_b @ A ) @ ( pow_set_b @ B ) ) ) ).

% Pow_mono
thf(fact_922_Pow__mono,axiom,
    ! [A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ord_le3795704787696855135_set_b @ ( pow_b @ A ) @ ( pow_b @ B ) ) ) ).

% Pow_mono
thf(fact_923_sigma__sets__mono_H_H,axiom,
    ! [A: set_d,C: set_d,D2: set_set_d,B: set_set_d] :
      ( ( member_set_d @ A @ ( sigma_sigma_sets_d @ C @ D2 ) )
     => ( ( ord_le3937772867789547105_set_d @ B @ D2 )
       => ( ( ord_le3937772867789547105_set_d @ D2 @ ( pow_d @ C ) )
         => ( ord_le3937772867789547105_set_d @ ( sigma_sigma_sets_d @ A @ B ) @ ( sigma_sigma_sets_d @ C @ D2 ) ) ) ) ) ).

% sigma_sets_mono''
thf(fact_924_sigma__sets__mono_H_H,axiom,
    ! [A: set_b,C: set_b,D2: set_set_b,B: set_set_b] :
      ( ( member_set_b @ A @ ( sigma_sigma_sets_b @ C @ D2 ) )
     => ( ( ord_le3795704787696855135_set_b @ B @ D2 )
       => ( ( ord_le3795704787696855135_set_b @ D2 @ ( pow_b @ C ) )
         => ( ord_le3795704787696855135_set_b @ ( sigma_sigma_sets_b @ A @ B ) @ ( sigma_sigma_sets_b @ C @ D2 ) ) ) ) ) ).

% sigma_sets_mono''
thf(fact_925_sigma__sets__sigma__sets__eq,axiom,
    ! [M: set_set_b,S: set_b] :
      ( ( ord_le3795704787696855135_set_b @ M @ ( pow_b @ S ) )
     => ( ( sigma_sigma_sets_b @ S @ ( sigma_sigma_sets_b @ S @ M ) )
        = ( sigma_sigma_sets_b @ S @ M ) ) ) ).

% sigma_sets_sigma_sets_eq
thf(fact_926_sets_Ospace__closed,axiom,
    ! [M: sigma_measure_d] : ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M ) @ ( pow_d @ ( sigma_space_d @ M ) ) ) ).

% sets.space_closed
thf(fact_927_sets_Ospace__closed,axiom,
    ! [M: sigma_measure_b] : ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M ) @ ( pow_b @ ( sigma_space_b @ M ) ) ) ).

% sets.space_closed
thf(fact_928_sigma__sets__into__sp,axiom,
    ! [A: set_set_d,Sp: set_d,X: set_d] :
      ( ( ord_le3937772867789547105_set_d @ A @ ( pow_d @ Sp ) )
     => ( ( member_set_d @ X @ ( sigma_sigma_sets_d @ Sp @ A ) )
       => ( ord_less_eq_set_d @ X @ Sp ) ) ) ).

% sigma_sets_into_sp
thf(fact_929_sigma__sets__into__sp,axiom,
    ! [A: set_set_set_b,Sp: set_set_b,X: set_set_b] :
      ( ( ord_le3201067847557142847_set_b @ A @ ( pow_set_b @ Sp ) )
     => ( ( member_set_set_b @ X @ ( sigma_2987359972167793591_set_b @ Sp @ A ) )
       => ( ord_le3795704787696855135_set_b @ X @ Sp ) ) ) ).

% sigma_sets_into_sp
thf(fact_930_sigma__sets__into__sp,axiom,
    ! [A: set_set_b,Sp: set_b,X: set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( pow_b @ Sp ) )
     => ( ( member_set_b @ X @ ( sigma_sigma_sets_b @ Sp @ A ) )
       => ( ord_less_eq_set_b @ X @ Sp ) ) ) ).

% sigma_sets_into_sp
thf(fact_931_Pi__cong,axiom,
    ! [A: set_b,F: b > d,G: b > d,B: b > set_d] :
      ( ! [W: b] :
          ( ( member_b @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_b_d @ F @ ( pi_b_d @ A @ B ) )
        = ( member_b_d @ G @ ( pi_b_d @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_932_Pi__cong,axiom,
    ! [A: set_b,F: b > b,G: b > b,B: b > set_b] :
      ( ! [W: b] :
          ( ( member_b @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_b_b @ F @ ( pi_b_b @ A @ B ) )
        = ( member_b_b @ G @ ( pi_b_b @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_933_Pi__cong,axiom,
    ! [A: set_d,F: d > b,G: d > b,B: d > set_b] :
      ( ! [W: d] :
          ( ( member_d @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_d_b @ F @ ( pi_d_b @ A @ B ) )
        = ( member_d_b @ G @ ( pi_d_b @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_934_Pi__cong,axiom,
    ! [A: set_d,F: d > d,G: d > d,B: d > set_d] :
      ( ! [W: d] :
          ( ( member_d @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_d_d @ F @ ( pi_d_d @ A @ B ) )
        = ( member_d_d @ G @ ( pi_d_d @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_935_Pi__mem,axiom,
    ! [F: c > c,A: set_c,B: c > set_c,X: c] :
      ( ( member_c_c @ F @ ( pi_c_c @ A @ B ) )
     => ( ( member_c @ X @ A )
       => ( member_c @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_936_Pi__mem,axiom,
    ! [F: b > d,A: set_b,B: b > set_d,X: b] :
      ( ( member_b_d @ F @ ( pi_b_d @ A @ B ) )
     => ( ( member_b @ X @ A )
       => ( member_d @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_937_Pi__mem,axiom,
    ! [F: b > b,A: set_b,B: b > set_b,X: b] :
      ( ( member_b_b @ F @ ( pi_b_b @ A @ B ) )
     => ( ( member_b @ X @ A )
       => ( member_b @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_938_Pi__mem,axiom,
    ! [F: d > b,A: set_d,B: d > set_b,X: d] :
      ( ( member_d_b @ F @ ( pi_d_b @ A @ B ) )
     => ( ( member_d @ X @ A )
       => ( member_b @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_939_Pi__mem,axiom,
    ! [F: d > d,A: set_d,B: d > set_d,X: d] :
      ( ( member_d_d @ F @ ( pi_d_d @ A @ B ) )
     => ( ( member_d @ X @ A )
       => ( member_d @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_940_Pi__mem,axiom,
    ! [F: set_d > c,A: set_set_d,B: set_d > set_c,X: set_d] :
      ( ( member_set_d_c @ F @ ( pi_set_d_c @ A @ B ) )
     => ( ( member_set_d @ X @ A )
       => ( member_c @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_941_Pi__mem,axiom,
    ! [F: c > set_d,A: set_c,B: c > set_set_d,X: c] :
      ( ( member_c_set_d @ F @ ( pi_c_set_d @ A @ B ) )
     => ( ( member_c @ X @ A )
       => ( member_set_d @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_942_Pi__mem,axiom,
    ! [F: c > set_b,A: set_c,B: c > set_set_b,X: c] :
      ( ( member_c_set_b @ F @ ( pi_c_set_b @ A @ B ) )
     => ( ( member_c @ X @ A )
       => ( member_set_b @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_943_Pi__mem,axiom,
    ! [F: set_b > c,A: set_set_b,B: set_b > set_c,X: set_b] :
      ( ( member_set_b_c @ F @ ( pi_set_b_c @ A @ B ) )
     => ( ( member_set_b @ X @ A )
       => ( member_c @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_944_Pi__mem,axiom,
    ! [F: set_d > set_d,A: set_set_d,B: set_d > set_set_d,X: set_d] :
      ( ( member_set_d_set_d @ F @ ( pi_set_d_set_d @ A @ B ) )
     => ( ( member_set_d @ X @ A )
       => ( member_set_d @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_945_Pi__iff,axiom,
    ! [F: b > d,I2: set_b,X4: b > set_d] :
      ( ( member_b_d @ F @ ( pi_b_d @ I2 @ X4 ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ I2 )
           => ( member_d @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_946_Pi__iff,axiom,
    ! [F: b > b,I2: set_b,X4: b > set_b] :
      ( ( member_b_b @ F @ ( pi_b_b @ I2 @ X4 ) )
      = ( ! [X2: b] :
            ( ( member_b @ X2 @ I2 )
           => ( member_b @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_947_Pi__iff,axiom,
    ! [F: d > b,I2: set_d,X4: d > set_b] :
      ( ( member_d_b @ F @ ( pi_d_b @ I2 @ X4 ) )
      = ( ! [X2: d] :
            ( ( member_d @ X2 @ I2 )
           => ( member_b @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_948_Pi__iff,axiom,
    ! [F: d > d,I2: set_d,X4: d > set_d] :
      ( ( member_d_d @ F @ ( pi_d_d @ I2 @ X4 ) )
      = ( ! [X2: d] :
            ( ( member_d @ X2 @ I2 )
           => ( member_d @ ( F @ X2 ) @ ( X4 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_949_Pi__I_H,axiom,
    ! [A: set_b,F: b > d,B: b > set_d] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_d @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_b_d @ F @ ( pi_b_d @ A @ B ) ) ) ).

% Pi_I'
thf(fact_950_Pi__I_H,axiom,
    ! [A: set_b,F: b > b,B: b > set_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_b_b @ F @ ( pi_b_b @ A @ B ) ) ) ).

% Pi_I'
thf(fact_951_Pi__I_H,axiom,
    ! [A: set_d,F: d > b,B: d > set_b] :
      ( ! [X3: d] :
          ( ( member_d @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_d_b @ F @ ( pi_d_b @ A @ B ) ) ) ).

% Pi_I'
thf(fact_952_Pi__I_H,axiom,
    ! [A: set_d,F: d > d,B: d > set_d] :
      ( ! [X3: d] :
          ( ( member_d @ X3 @ A )
         => ( member_d @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_d_d @ F @ ( pi_d_d @ A @ B ) ) ) ).

% Pi_I'
thf(fact_953_Pi__I_H,axiom,
    ! [A: set_c,F: c > c,B: c > set_c] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_c @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_c_c @ F @ ( pi_c_c @ A @ B ) ) ) ).

% Pi_I'
thf(fact_954_Pi__I_H,axiom,
    ! [A: set_set_d,F: set_d > c,B: set_d > set_c] :
      ( ! [X3: set_d] :
          ( ( member_set_d @ X3 @ A )
         => ( member_c @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_set_d_c @ F @ ( pi_set_d_c @ A @ B ) ) ) ).

% Pi_I'
thf(fact_955_Pi__I_H,axiom,
    ! [A: set_c,F: c > set_d,B: c > set_set_d] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_set_d @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_c_set_d @ F @ ( pi_c_set_d @ A @ B ) ) ) ).

% Pi_I'
thf(fact_956_Pi__I_H,axiom,
    ! [A: set_c,F: c > set_b,B: c > set_set_b] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_set_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_c_set_b @ F @ ( pi_c_set_b @ A @ B ) ) ) ).

% Pi_I'
thf(fact_957_Pi__I_H,axiom,
    ! [A: set_set_b,F: set_b > c,B: set_b > set_c] :
      ( ! [X3: set_b] :
          ( ( member_set_b @ X3 @ A )
         => ( member_c @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_set_b_c @ F @ ( pi_set_b_c @ A @ B ) ) ) ).

% Pi_I'
thf(fact_958_Pi__I_H,axiom,
    ! [A: set_set_d,F: set_d > set_d,B: set_d > set_set_d] :
      ( ! [X3: set_d] :
          ( ( member_set_d @ X3 @ A )
         => ( member_set_d @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_set_d_set_d @ F @ ( pi_set_d_set_d @ A @ B ) ) ) ).

% Pi_I'
thf(fact_959_PiE,axiom,
    ! [F: c > c,A: set_c,B: c > set_c,X: c] :
      ( ( member_c_c @ F @ ( pi_c_c @ A @ B ) )
     => ( ~ ( member_c @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_c @ X @ A ) ) ) ).

% PiE
thf(fact_960_PiE,axiom,
    ! [F: b > d,A: set_b,B: b > set_d,X: b] :
      ( ( member_b_d @ F @ ( pi_b_d @ A @ B ) )
     => ( ~ ( member_d @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_b @ X @ A ) ) ) ).

% PiE
thf(fact_961_PiE,axiom,
    ! [F: b > b,A: set_b,B: b > set_b,X: b] :
      ( ( member_b_b @ F @ ( pi_b_b @ A @ B ) )
     => ( ~ ( member_b @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_b @ X @ A ) ) ) ).

% PiE
thf(fact_962_PiE,axiom,
    ! [F: d > b,A: set_d,B: d > set_b,X: d] :
      ( ( member_d_b @ F @ ( pi_d_b @ A @ B ) )
     => ( ~ ( member_b @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_d @ X @ A ) ) ) ).

% PiE
thf(fact_963_PiE,axiom,
    ! [F: d > d,A: set_d,B: d > set_d,X: d] :
      ( ( member_d_d @ F @ ( pi_d_d @ A @ B ) )
     => ( ~ ( member_d @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_d @ X @ A ) ) ) ).

% PiE
thf(fact_964_PiE,axiom,
    ! [F: c > set_d,A: set_c,B: c > set_set_d,X: c] :
      ( ( member_c_set_d @ F @ ( pi_c_set_d @ A @ B ) )
     => ( ~ ( member_set_d @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_c @ X @ A ) ) ) ).

% PiE
thf(fact_965_PiE,axiom,
    ! [F: set_d > c,A: set_set_d,B: set_d > set_c,X: set_d] :
      ( ( member_set_d_c @ F @ ( pi_set_d_c @ A @ B ) )
     => ( ~ ( member_c @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_set_d @ X @ A ) ) ) ).

% PiE
thf(fact_966_PiE,axiom,
    ! [F: set_b > c,A: set_set_b,B: set_b > set_c,X: set_b] :
      ( ( member_set_b_c @ F @ ( pi_set_b_c @ A @ B ) )
     => ( ~ ( member_c @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_set_b @ X @ A ) ) ) ).

% PiE
thf(fact_967_PiE,axiom,
    ! [F: c > set_b,A: set_c,B: c > set_set_b,X: c] :
      ( ( member_c_set_b @ F @ ( pi_c_set_b @ A @ B ) )
     => ( ~ ( member_set_b @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_c @ X @ A ) ) ) ).

% PiE
thf(fact_968_PiE,axiom,
    ! [F: set_d > set_d,A: set_set_d,B: set_d > set_set_d,X: set_d] :
      ( ( member_set_d_set_d @ F @ ( pi_set_d_set_d @ A @ B ) )
     => ( ~ ( member_set_d @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_set_d @ X @ A ) ) ) ).

% PiE
thf(fact_969_funcset__mem,axiom,
    ! [F: c > c,A: set_c,B: set_c,X: c] :
      ( ( member_c_c @ F
        @ ( pi_c_c @ A
          @ ^ [Uu: c] : B ) )
     => ( ( member_c @ X @ A )
       => ( member_c @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_970_funcset__mem,axiom,
    ! [F: b > d,A: set_b,B: set_d,X: b] :
      ( ( member_b_d @ F
        @ ( pi_b_d @ A
          @ ^ [Uu: b] : B ) )
     => ( ( member_b @ X @ A )
       => ( member_d @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_971_funcset__mem,axiom,
    ! [F: b > b,A: set_b,B: set_b,X: b] :
      ( ( member_b_b @ F
        @ ( pi_b_b @ A
          @ ^ [Uu: b] : B ) )
     => ( ( member_b @ X @ A )
       => ( member_b @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_972_funcset__mem,axiom,
    ! [F: d > b,A: set_d,B: set_b,X: d] :
      ( ( member_d_b @ F
        @ ( pi_d_b @ A
          @ ^ [Uu: d] : B ) )
     => ( ( member_d @ X @ A )
       => ( member_b @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_973_funcset__mem,axiom,
    ! [F: d > d,A: set_d,B: set_d,X: d] :
      ( ( member_d_d @ F
        @ ( pi_d_d @ A
          @ ^ [Uu: d] : B ) )
     => ( ( member_d @ X @ A )
       => ( member_d @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_974_funcset__mem,axiom,
    ! [F: set_d > c,A: set_set_d,B: set_c,X: set_d] :
      ( ( member_set_d_c @ F
        @ ( pi_set_d_c @ A
          @ ^ [Uu: set_d] : B ) )
     => ( ( member_set_d @ X @ A )
       => ( member_c @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_975_funcset__mem,axiom,
    ! [F: c > set_d,A: set_c,B: set_set_d,X: c] :
      ( ( member_c_set_d @ F
        @ ( pi_c_set_d @ A
          @ ^ [Uu: c] : B ) )
     => ( ( member_c @ X @ A )
       => ( member_set_d @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_976_funcset__mem,axiom,
    ! [F: c > set_b,A: set_c,B: set_set_b,X: c] :
      ( ( member_c_set_b @ F
        @ ( pi_c_set_b @ A
          @ ^ [Uu: c] : B ) )
     => ( ( member_c @ X @ A )
       => ( member_set_b @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_977_funcset__mem,axiom,
    ! [F: set_b > c,A: set_set_b,B: set_c,X: set_b] :
      ( ( member_set_b_c @ F
        @ ( pi_set_b_c @ A
          @ ^ [Uu: set_b] : B ) )
     => ( ( member_set_b @ X @ A )
       => ( member_c @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_978_funcset__mem,axiom,
    ! [F: set_d > set_d,A: set_set_d,B: set_set_d,X: set_d] :
      ( ( member_set_d_set_d @ F
        @ ( pi_set_d_set_d @ A
          @ ^ [Uu: set_d] : B ) )
     => ( ( member_set_d @ X @ A )
       => ( member_set_d @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_979_funcset__id,axiom,
    ! [A: set_b] :
      ( member_b_b
      @ ^ [X2: b] : X2
      @ ( pi_b_b @ A
        @ ^ [Uu: b] : A ) ) ).

% funcset_id
thf(fact_980_funcset__id,axiom,
    ! [A: set_d] :
      ( member_d_d
      @ ^ [X2: d] : X2
      @ ( pi_d_d @ A
        @ ^ [Uu: d] : A ) ) ).

% funcset_id
thf(fact_981_funcsetI,axiom,
    ! [A: set_b,F: b > d,B: set_d] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_d @ ( F @ X3 ) @ B ) )
     => ( member_b_d @ F
        @ ( pi_b_d @ A
          @ ^ [Uu: b] : B ) ) ) ).

% funcsetI
thf(fact_982_funcsetI,axiom,
    ! [A: set_b,F: b > b,B: set_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ B ) )
     => ( member_b_b @ F
        @ ( pi_b_b @ A
          @ ^ [Uu: b] : B ) ) ) ).

% funcsetI
thf(fact_983_funcsetI,axiom,
    ! [A: set_d,F: d > b,B: set_b] :
      ( ! [X3: d] :
          ( ( member_d @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ B ) )
     => ( member_d_b @ F
        @ ( pi_d_b @ A
          @ ^ [Uu: d] : B ) ) ) ).

% funcsetI
thf(fact_984_funcsetI,axiom,
    ! [A: set_d,F: d > d,B: set_d] :
      ( ! [X3: d] :
          ( ( member_d @ X3 @ A )
         => ( member_d @ ( F @ X3 ) @ B ) )
     => ( member_d_d @ F
        @ ( pi_d_d @ A
          @ ^ [Uu: d] : B ) ) ) ).

% funcsetI
thf(fact_985_funcsetI,axiom,
    ! [A: set_c,F: c > c,B: set_c] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_c @ ( F @ X3 ) @ B ) )
     => ( member_c_c @ F
        @ ( pi_c_c @ A
          @ ^ [Uu: c] : B ) ) ) ).

% funcsetI
thf(fact_986_funcsetI,axiom,
    ! [A: set_set_d,F: set_d > c,B: set_c] :
      ( ! [X3: set_d] :
          ( ( member_set_d @ X3 @ A )
         => ( member_c @ ( F @ X3 ) @ B ) )
     => ( member_set_d_c @ F
        @ ( pi_set_d_c @ A
          @ ^ [Uu: set_d] : B ) ) ) ).

% funcsetI
thf(fact_987_funcsetI,axiom,
    ! [A: set_c,F: c > set_d,B: set_set_d] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_set_d @ ( F @ X3 ) @ B ) )
     => ( member_c_set_d @ F
        @ ( pi_c_set_d @ A
          @ ^ [Uu: c] : B ) ) ) ).

% funcsetI
thf(fact_988_funcsetI,axiom,
    ! [A: set_c,F: c > set_b,B: set_set_b] :
      ( ! [X3: c] :
          ( ( member_c @ X3 @ A )
         => ( member_set_b @ ( F @ X3 ) @ B ) )
     => ( member_c_set_b @ F
        @ ( pi_c_set_b @ A
          @ ^ [Uu: c] : B ) ) ) ).

% funcsetI
thf(fact_989_funcsetI,axiom,
    ! [A: set_set_b,F: set_b > c,B: set_c] :
      ( ! [X3: set_b] :
          ( ( member_set_b @ X3 @ A )
         => ( member_c @ ( F @ X3 ) @ B ) )
     => ( member_set_b_c @ F
        @ ( pi_set_b_c @ A
          @ ^ [Uu: set_b] : B ) ) ) ).

% funcsetI
thf(fact_990_funcsetI,axiom,
    ! [A: set_set_d,F: set_d > set_d,B: set_set_d] :
      ( ! [X3: set_d] :
          ( ( member_set_d @ X3 @ A )
         => ( member_set_d @ ( F @ X3 ) @ B ) )
     => ( member_set_d_set_d @ F
        @ ( pi_set_d_set_d @ A
          @ ^ [Uu: set_d] : B ) ) ) ).

% funcsetI
thf(fact_991_measurable__measure__of,axiom,
    ! [N: set_set_d,Omega: set_d,F: set_b > d,M: sigma_measure_set_b,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ N @ ( pow_d @ Omega ) )
     => ( ( member_set_b_d @ F
          @ ( pi_set_b_d @ ( sigma_space_set_b @ M )
            @ ^ [Uu: set_b] : Omega ) )
       => ( ! [Y5: set_d] :
              ( ( member_set_d @ Y5 @ N )
             => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ Y5 ) @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) ) )
         => ( member_set_b_d @ F @ ( sigma_3265996224607154482et_b_d @ M @ ( sigma_measure_of_d @ Omega @ N @ Mu2 ) ) ) ) ) ) ).

% measurable_measure_of
thf(fact_992_measurable__measure__of,axiom,
    ! [N: set_set_d,Omega: set_d,F: b > d,M: sigma_measure_b,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ N @ ( pow_d @ Omega ) )
     => ( ( member_b_d @ F
          @ ( pi_b_d @ ( sigma_space_b @ M )
            @ ^ [Uu: b] : Omega ) )
       => ( ! [Y5: set_d] :
              ( ( member_set_d @ Y5 @ N )
             => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_d @ F @ Y5 ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) )
         => ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ ( sigma_measure_of_d @ Omega @ N @ Mu2 ) ) ) ) ) ) ).

% measurable_measure_of
thf(fact_993_measurable__measure__of,axiom,
    ! [N: set_set_d,Omega: set_d,F: d > d,M: sigma_measure_d,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ N @ ( pow_d @ Omega ) )
     => ( ( member_d_d @ F
          @ ( pi_d_d @ ( sigma_space_d @ M )
            @ ^ [Uu: d] : Omega ) )
       => ( ! [Y5: set_d] :
              ( ( member_set_d @ Y5 @ N )
             => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_d @ F @ Y5 ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) )
         => ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ ( sigma_measure_of_d @ Omega @ N @ Mu2 ) ) ) ) ) ) ).

% measurable_measure_of
thf(fact_994_measurable__measure__of,axiom,
    ! [N: set_set_b,Omega: set_b,F: set_b > b,M: sigma_measure_set_b,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ N @ ( pow_b @ Omega ) )
     => ( ( member_set_b_b @ F
          @ ( pi_set_b_b @ ( sigma_space_set_b @ M )
            @ ^ [Uu: set_b] : Omega ) )
       => ( ! [Y5: set_b] :
              ( ( member_set_b @ Y5 @ N )
             => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ Y5 ) @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) ) )
         => ( member_set_b_b @ F @ ( sigma_3265996224607154480et_b_b @ M @ ( sigma_measure_of_b @ Omega @ N @ Mu2 ) ) ) ) ) ) ).

% measurable_measure_of
thf(fact_995_measurable__measure__of,axiom,
    ! [N: set_set_b,Omega: set_b,F: b > b,M: sigma_measure_b,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ N @ ( pow_b @ Omega ) )
     => ( ( member_b_b @ F
          @ ( pi_b_b @ ( sigma_space_b @ M )
            @ ^ [Uu: b] : Omega ) )
       => ( ! [Y5: set_b] :
              ( ( member_set_b @ Y5 @ N )
             => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_b @ F @ Y5 ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) )
         => ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ ( sigma_measure_of_b @ Omega @ N @ Mu2 ) ) ) ) ) ) ).

% measurable_measure_of
thf(fact_996_measurable__measure__of,axiom,
    ! [N: set_set_b,Omega: set_b,F: d > b,M: sigma_measure_d,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ N @ ( pow_b @ Omega ) )
     => ( ( member_d_b @ F
          @ ( pi_d_b @ ( sigma_space_d @ M )
            @ ^ [Uu: d] : Omega ) )
       => ( ! [Y5: set_b] :
              ( ( member_set_b @ Y5 @ N )
             => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_b @ F @ Y5 ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) )
         => ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ ( sigma_measure_of_b @ Omega @ N @ Mu2 ) ) ) ) ) ) ).

% measurable_measure_of
thf(fact_997_measurable__iff__measure__of,axiom,
    ! [N: set_set_d,Omega: set_d,F: b > d,M: sigma_measure_b,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ N @ ( pow_d @ Omega ) )
     => ( ( member_b_d @ F
          @ ( pi_b_d @ ( sigma_space_b @ M )
            @ ^ [Uu: b] : Omega ) )
       => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ ( sigma_measure_of_d @ Omega @ N @ Mu2 ) ) )
          = ( ! [X2: set_d] :
                ( ( member_set_d @ X2 @ N )
               => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_d @ F @ X2 ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) ) ) ) ) ) ).

% measurable_iff_measure_of
thf(fact_998_measurable__iff__measure__of,axiom,
    ! [N: set_set_d,Omega: set_d,F: d > d,M: sigma_measure_d,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ N @ ( pow_d @ Omega ) )
     => ( ( member_d_d @ F
          @ ( pi_d_d @ ( sigma_space_d @ M )
            @ ^ [Uu: d] : Omega ) )
       => ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ ( sigma_measure_of_d @ Omega @ N @ Mu2 ) ) )
          = ( ! [X2: set_d] :
                ( ( member_set_d @ X2 @ N )
               => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_d @ F @ X2 ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) ) ) ) ) ) ).

% measurable_iff_measure_of
thf(fact_999_measurable__iff__measure__of,axiom,
    ! [N: set_set_b,Omega: set_b,F: set_b > b,M: sigma_measure_set_b,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ N @ ( pow_b @ Omega ) )
     => ( ( member_set_b_b @ F
          @ ( pi_set_b_b @ ( sigma_space_set_b @ M )
            @ ^ [Uu: set_b] : Omega ) )
       => ( ( member_set_b_b @ F @ ( sigma_3265996224607154480et_b_b @ M @ ( sigma_measure_of_b @ Omega @ N @ Mu2 ) ) )
          = ( ! [X2: set_b] :
                ( ( member_set_b @ X2 @ N )
               => ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ X2 ) @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) ) ) ) ) ) ) ).

% measurable_iff_measure_of
thf(fact_1000_measurable__iff__measure__of,axiom,
    ! [N: set_set_b,Omega: set_b,F: b > b,M: sigma_measure_b,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ N @ ( pow_b @ Omega ) )
     => ( ( member_b_b @ F
          @ ( pi_b_b @ ( sigma_space_b @ M )
            @ ^ [Uu: b] : Omega ) )
       => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ ( sigma_measure_of_b @ Omega @ N @ Mu2 ) ) )
          = ( ! [X2: set_b] :
                ( ( member_set_b @ X2 @ N )
               => ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_b @ F @ X2 ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) ) ) ) ) ) ).

% measurable_iff_measure_of
thf(fact_1001_measurable__iff__measure__of,axiom,
    ! [N: set_set_b,Omega: set_b,F: d > b,M: sigma_measure_d,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ N @ ( pow_b @ Omega ) )
     => ( ( member_d_b @ F
          @ ( pi_d_b @ ( sigma_space_d @ M )
            @ ^ [Uu: d] : Omega ) )
       => ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ ( sigma_measure_of_b @ Omega @ N @ Mu2 ) ) )
          = ( ! [X2: set_b] :
                ( ( member_set_b @ X2 @ N )
               => ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_b @ F @ X2 ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) ) ) ) ) ) ).

% measurable_iff_measure_of
thf(fact_1002_Ball__Collect,axiom,
    ( ball_set_b
    = ( ^ [A3: set_set_b,P2: set_b > $o] : ( ord_le3795704787696855135_set_b @ A3 @ ( collect_set_b @ P2 ) ) ) ) ).

% Ball_Collect
thf(fact_1003_Ball__Collect,axiom,
    ( ball_b
    = ( ^ [A3: set_b,P2: b > $o] : ( ord_less_eq_set_b @ A3 @ ( collect_b @ P2 ) ) ) ) ).

% Ball_Collect
thf(fact_1004_restrict__distr,axiom,
    ! [F: b > set_b,M: sigma_measure_b,N: sigma_measure_set_b,Omega: set_set_b] :
      ( ( member_b_set_b @ F @ ( sigma_5029598207865514736_set_b @ M @ N ) )
     => ( ( member_set_set_b @ ( inf_inf_set_set_b @ Omega @ ( sigma_space_set_b @ N ) ) @ ( sigma_sets_set_b @ N ) )
       => ( ( member_b_set_b @ F
            @ ( pi_b_set_b @ ( sigma_space_b @ M )
              @ ^ [Uu: b] : Omega ) )
         => ( ( sigma_5799942775684512035_set_b @ ( measur8192432674156524440_set_b @ M @ N @ F ) @ Omega )
            = ( measur8192432674156524440_set_b @ M @ ( sigma_5799942775684512035_set_b @ N @ Omega ) @ F ) ) ) ) ) ).

% restrict_distr
thf(fact_1005_restrict__distr,axiom,
    ! [F: d > set_b,M: sigma_measure_d,N: sigma_measure_set_b,Omega: set_set_b] :
      ( ( member_d_set_b @ F @ ( sigma_7718528281484490606_set_b @ M @ N ) )
     => ( ( member_set_set_b @ ( inf_inf_set_set_b @ Omega @ ( sigma_space_set_b @ N ) ) @ ( sigma_sets_set_b @ N ) )
       => ( ( member_d_set_b @ F
            @ ( pi_d_set_b @ ( sigma_space_d @ M )
              @ ^ [Uu: d] : Omega ) )
         => ( ( sigma_5799942775684512035_set_b @ ( measur1657990710920724502_set_b @ M @ N @ F ) @ Omega )
            = ( measur1657990710920724502_set_b @ M @ ( sigma_5799942775684512035_set_b @ N @ Omega ) @ F ) ) ) ) ) ).

% restrict_distr
thf(fact_1006_restrict__distr,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d,Omega: set_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
     => ( ( member_set_d @ ( inf_inf_set_d @ Omega @ ( sigma_space_d @ N ) ) @ ( sigma_sets_d @ N ) )
       => ( ( member_b_d @ F
            @ ( pi_b_d @ ( sigma_space_b @ M )
              @ ^ [Uu: b] : Omega ) )
         => ( ( sigma_8692839461743104069pace_d @ ( measure_distr_b_d @ M @ N @ F ) @ Omega )
            = ( measure_distr_b_d @ M @ ( sigma_8692839461743104069pace_d @ N @ Omega ) @ F ) ) ) ) ) ).

% restrict_distr
thf(fact_1007_restrict__distr,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,Omega: set_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( ( member_set_b @ ( inf_inf_set_b @ Omega @ ( sigma_space_b @ N ) ) @ ( sigma_sets_b @ N ) )
       => ( ( member_b_b @ F
            @ ( pi_b_b @ ( sigma_space_b @ M )
              @ ^ [Uu: b] : Omega ) )
         => ( ( sigma_8692839461743104067pace_b @ ( measure_distr_b_b @ M @ N @ F ) @ Omega )
            = ( measure_distr_b_b @ M @ ( sigma_8692839461743104067pace_b @ N @ Omega ) @ F ) ) ) ) ) ).

% restrict_distr
thf(fact_1008_restrict__distr,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b,Omega: set_b] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
     => ( ( member_set_b @ ( inf_inf_set_b @ Omega @ ( sigma_space_b @ N ) ) @ ( sigma_sets_b @ N ) )
       => ( ( member_d_b @ F
            @ ( pi_d_b @ ( sigma_space_d @ M )
              @ ^ [Uu: d] : Omega ) )
         => ( ( sigma_8692839461743104067pace_b @ ( measure_distr_d_b @ M @ N @ F ) @ Omega )
            = ( measure_distr_d_b @ M @ ( sigma_8692839461743104067pace_b @ N @ Omega ) @ F ) ) ) ) ) ).

% restrict_distr
thf(fact_1009_restrict__distr,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d,Omega: set_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
     => ( ( member_set_d @ ( inf_inf_set_d @ Omega @ ( sigma_space_d @ N ) ) @ ( sigma_sets_d @ N ) )
       => ( ( member_d_d @ F
            @ ( pi_d_d @ ( sigma_space_d @ M )
              @ ^ [Uu: d] : Omega ) )
         => ( ( sigma_8692839461743104069pace_d @ ( measure_distr_d_d @ M @ N @ F ) @ Omega )
            = ( measure_distr_d_d @ M @ ( sigma_8692839461743104069pace_d @ N @ Omega ) @ F ) ) ) ) ) ).

% restrict_distr
thf(fact_1010_measure__distr,axiom,
    ! [F: set_b > d,M: sigma_measure_set_b,N: sigma_measure_d,S: set_d] :
      ( ( member_set_b_d @ F @ ( sigma_3265996224607154482et_b_d @ M @ N ) )
     => ( ( member_set_d @ S @ ( sigma_sets_d @ N ) )
       => ( ( sigma_measure_d2 @ ( measur6428830690898164186et_b_d @ M @ N @ F ) @ S )
          = ( sigma_measure_set_b2 @ M @ ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ S ) @ ( sigma_space_set_b @ M ) ) ) ) ) ) ).

% measure_distr
thf(fact_1011_measure__distr,axiom,
    ! [F: set_b > b,M: sigma_measure_set_b,N: sigma_measure_b,S: set_b] :
      ( ( member_set_b_b @ F @ ( sigma_3265996224607154480et_b_b @ M @ N ) )
     => ( ( member_set_b @ S @ ( sigma_sets_b @ N ) )
       => ( ( sigma_measure_b2 @ ( measur6428830690898164184et_b_b @ M @ N @ F ) @ S )
          = ( sigma_measure_set_b2 @ M @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ S ) @ ( sigma_space_set_b @ M ) ) ) ) ) ) ).

% measure_distr
thf(fact_1012_measure__distr,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d,S: set_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
     => ( ( member_set_d @ S @ ( sigma_sets_d @ N ) )
       => ( ( sigma_measure_d2 @ ( measure_distr_b_d @ M @ N @ F ) @ S )
          = ( sigma_measure_b2 @ M @ ( inf_inf_set_b @ ( vimage_b_d @ F @ S ) @ ( sigma_space_b @ M ) ) ) ) ) ) ).

% measure_distr
thf(fact_1013_measure__distr,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,S: set_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( ( member_set_b @ S @ ( sigma_sets_b @ N ) )
       => ( ( sigma_measure_b2 @ ( measure_distr_b_b @ M @ N @ F ) @ S )
          = ( sigma_measure_b2 @ M @ ( inf_inf_set_b @ ( vimage_b_b @ F @ S ) @ ( sigma_space_b @ M ) ) ) ) ) ) ).

% measure_distr
thf(fact_1014_measure__distr,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b,S: set_b] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
     => ( ( member_set_b @ S @ ( sigma_sets_b @ N ) )
       => ( ( sigma_measure_b2 @ ( measure_distr_d_b @ M @ N @ F ) @ S )
          = ( sigma_measure_d2 @ M @ ( inf_inf_set_d @ ( vimage_d_b @ F @ S ) @ ( sigma_space_d @ M ) ) ) ) ) ) ).

% measure_distr
thf(fact_1015_measure__distr,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d,S: set_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
     => ( ( member_set_d @ S @ ( sigma_sets_d @ N ) )
       => ( ( sigma_measure_d2 @ ( measure_distr_d_d @ M @ N @ F ) @ S )
          = ( sigma_measure_d2 @ M @ ( inf_inf_set_d @ ( vimage_d_d @ F @ S ) @ ( sigma_space_d @ M ) ) ) ) ) ) ).

% measure_distr
thf(fact_1016_null__sets__distr__iff,axiom,
    ! [F: set_b > d,M: sigma_measure_set_b,N: sigma_measure_d,A: set_d] :
      ( ( member_set_b_d @ F @ ( sigma_3265996224607154482et_b_d @ M @ N ) )
     => ( ( member_set_d @ A @ ( measure_null_sets_d @ ( measur6428830690898164186et_b_d @ M @ N @ F ) ) )
        = ( ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ A ) @ ( sigma_space_set_b @ M ) ) @ ( measur1516554132335629586_set_b @ M ) )
          & ( member_set_d @ A @ ( sigma_sets_d @ N ) ) ) ) ) ).

% null_sets_distr_iff
thf(fact_1017_null__sets__distr__iff,axiom,
    ! [F: set_b > b,M: sigma_measure_set_b,N: sigma_measure_b,A: set_b] :
      ( ( member_set_b_b @ F @ ( sigma_3265996224607154480et_b_b @ M @ N ) )
     => ( ( member_set_b @ A @ ( measure_null_sets_b @ ( measur6428830690898164184et_b_b @ M @ N @ F ) ) )
        = ( ( member_set_set_b @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ A ) @ ( sigma_space_set_b @ M ) ) @ ( measur1516554132335629586_set_b @ M ) )
          & ( member_set_b @ A @ ( sigma_sets_b @ N ) ) ) ) ) ).

% null_sets_distr_iff
thf(fact_1018_null__sets__distr__iff,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d,A: set_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
     => ( ( member_set_d @ A @ ( measure_null_sets_d @ ( measure_distr_b_d @ M @ N @ F ) ) )
        = ( ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_d @ F @ A ) @ ( sigma_space_b @ M ) ) @ ( measure_null_sets_b @ M ) )
          & ( member_set_d @ A @ ( sigma_sets_d @ N ) ) ) ) ) ).

% null_sets_distr_iff
thf(fact_1019_null__sets__distr__iff,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,A: set_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( ( member_set_b @ A @ ( measure_null_sets_b @ ( measure_distr_b_b @ M @ N @ F ) ) )
        = ( ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_b @ F @ A ) @ ( sigma_space_b @ M ) ) @ ( measure_null_sets_b @ M ) )
          & ( member_set_b @ A @ ( sigma_sets_b @ N ) ) ) ) ) ).

% null_sets_distr_iff
thf(fact_1020_null__sets__distr__iff,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b,A: set_b] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
     => ( ( member_set_b @ A @ ( measure_null_sets_b @ ( measure_distr_d_b @ M @ N @ F ) ) )
        = ( ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_b @ F @ A ) @ ( sigma_space_d @ M ) ) @ ( measure_null_sets_d @ M ) )
          & ( member_set_b @ A @ ( sigma_sets_b @ N ) ) ) ) ) ).

% null_sets_distr_iff
thf(fact_1021_null__sets__distr__iff,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d,A: set_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
     => ( ( member_set_d @ A @ ( measure_null_sets_d @ ( measure_distr_d_d @ M @ N @ F ) ) )
        = ( ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_d @ F @ A ) @ ( sigma_space_d @ M ) ) @ ( measure_null_sets_d @ M ) )
          & ( member_set_d @ A @ ( sigma_sets_d @ N ) ) ) ) ) ).

% null_sets_distr_iff
thf(fact_1022_distr__def,axiom,
    ( measure_distr_b_b
    = ( ^ [M3: sigma_measure_b,N3: sigma_measure_b,F2: b > b] :
          ( sigma_measure_of_b @ ( sigma_space_b @ N3 ) @ ( sigma_sets_b @ N3 )
          @ ^ [A3: set_b] : ( sigma_emeasure_b @ M3 @ ( inf_inf_set_b @ ( vimage_b_b @ F2 @ A3 ) @ ( sigma_space_b @ M3 ) ) ) ) ) ) ).

% distr_def
thf(fact_1023_distr__def,axiom,
    ( measure_distr_b_d
    = ( ^ [M3: sigma_measure_b,N3: sigma_measure_d,F2: b > d] :
          ( sigma_measure_of_d @ ( sigma_space_d @ N3 ) @ ( sigma_sets_d @ N3 )
          @ ^ [A3: set_d] : ( sigma_emeasure_b @ M3 @ ( inf_inf_set_b @ ( vimage_b_d @ F2 @ A3 ) @ ( sigma_space_b @ M3 ) ) ) ) ) ) ).

% distr_def
thf(fact_1024_distr__def,axiom,
    ( measure_distr_d_b
    = ( ^ [M3: sigma_measure_d,N3: sigma_measure_b,F2: d > b] :
          ( sigma_measure_of_b @ ( sigma_space_b @ N3 ) @ ( sigma_sets_b @ N3 )
          @ ^ [A3: set_b] : ( sigma_emeasure_d @ M3 @ ( inf_inf_set_d @ ( vimage_d_b @ F2 @ A3 ) @ ( sigma_space_d @ M3 ) ) ) ) ) ) ).

% distr_def
thf(fact_1025_distr__def,axiom,
    ( measure_distr_d_d
    = ( ^ [M3: sigma_measure_d,N3: sigma_measure_d,F2: d > d] :
          ( sigma_measure_of_d @ ( sigma_space_d @ N3 ) @ ( sigma_sets_d @ N3 )
          @ ^ [A3: set_d] : ( sigma_emeasure_d @ M3 @ ( inf_inf_set_d @ ( vimage_d_d @ F2 @ A3 ) @ ( sigma_space_d @ M3 ) ) ) ) ) ) ).

% distr_def
thf(fact_1026_distr__def,axiom,
    ( measur6428830690898164184et_b_b
    = ( ^ [M3: sigma_measure_set_b,N3: sigma_measure_b,F2: set_b > b] :
          ( sigma_measure_of_b @ ( sigma_space_b @ N3 ) @ ( sigma_sets_b @ N3 )
          @ ^ [A3: set_b] : ( sigma_emeasure_set_b @ M3 @ ( inf_inf_set_set_b @ ( vimage_set_b_b @ F2 @ A3 ) @ ( sigma_space_set_b @ M3 ) ) ) ) ) ) ).

% distr_def
thf(fact_1027_distr__def,axiom,
    ( measur6428830690898164186et_b_d
    = ( ^ [M3: sigma_measure_set_b,N3: sigma_measure_d,F2: set_b > d] :
          ( sigma_measure_of_d @ ( sigma_space_d @ N3 ) @ ( sigma_sets_d @ N3 )
          @ ^ [A3: set_d] : ( sigma_emeasure_set_b @ M3 @ ( inf_inf_set_set_b @ ( vimage_set_b_d @ F2 @ A3 ) @ ( sigma_space_set_b @ M3 ) ) ) ) ) ) ).

% distr_def
thf(fact_1028_null__sets_OInt,axiom,
    ! [A2: set_b,M: sigma_measure_b,B2: set_b] :
      ( ( member_set_b @ A2 @ ( measure_null_sets_b @ M ) )
     => ( ( member_set_b @ B2 @ ( measure_null_sets_b @ M ) )
       => ( member_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( measure_null_sets_b @ M ) ) ) ) ).

% null_sets.Int
thf(fact_1029_null__sets_OInt,axiom,
    ! [A2: set_d,M: sigma_measure_d,B2: set_d] :
      ( ( member_set_d @ A2 @ ( measure_null_sets_d @ M ) )
     => ( ( member_set_d @ B2 @ ( measure_null_sets_d @ M ) )
       => ( member_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ ( measure_null_sets_d @ M ) ) ) ) ).

% null_sets.Int
thf(fact_1030_null__sets_OInt,axiom,
    ! [A2: set_set_b,M: sigma_measure_set_b,B2: set_set_b] :
      ( ( member_set_set_b @ A2 @ ( measur1516554132335629586_set_b @ M ) )
     => ( ( member_set_set_b @ B2 @ ( measur1516554132335629586_set_b @ M ) )
       => ( member_set_set_b @ ( inf_inf_set_set_b @ A2 @ B2 ) @ ( measur1516554132335629586_set_b @ M ) ) ) ) ).

% null_sets.Int
thf(fact_1031_null__sets_OInt__space__eq2,axiom,
    ! [X: set_b,M: sigma_measure_b] :
      ( ( member_set_b @ X @ ( measure_null_sets_b @ M ) )
     => ( ( inf_inf_set_b @ X @ ( sigma_space_b @ M ) )
        = X ) ) ).

% null_sets.Int_space_eq2
thf(fact_1032_null__sets_OInt__space__eq2,axiom,
    ! [X: set_d,M: sigma_measure_d] :
      ( ( member_set_d @ X @ ( measure_null_sets_d @ M ) )
     => ( ( inf_inf_set_d @ X @ ( sigma_space_d @ M ) )
        = X ) ) ).

% null_sets.Int_space_eq2
thf(fact_1033_null__sets_OInt__space__eq2,axiom,
    ! [X: set_set_b,M: sigma_measure_set_b] :
      ( ( member_set_set_b @ X @ ( measur1516554132335629586_set_b @ M ) )
     => ( ( inf_inf_set_set_b @ X @ ( sigma_space_set_b @ M ) )
        = X ) ) ).

% null_sets.Int_space_eq2
thf(fact_1034_null__sets_OInt__space__eq1,axiom,
    ! [X: set_b,M: sigma_measure_b] :
      ( ( member_set_b @ X @ ( measure_null_sets_b @ M ) )
     => ( ( inf_inf_set_b @ ( sigma_space_b @ M ) @ X )
        = X ) ) ).

% null_sets.Int_space_eq1
thf(fact_1035_null__sets_OInt__space__eq1,axiom,
    ! [X: set_d,M: sigma_measure_d] :
      ( ( member_set_d @ X @ ( measure_null_sets_d @ M ) )
     => ( ( inf_inf_set_d @ ( sigma_space_d @ M ) @ X )
        = X ) ) ).

% null_sets.Int_space_eq1
thf(fact_1036_null__sets_OInt__space__eq1,axiom,
    ! [X: set_set_b,M: sigma_measure_set_b] :
      ( ( member_set_set_b @ X @ ( measur1516554132335629586_set_b @ M ) )
     => ( ( inf_inf_set_set_b @ ( sigma_space_set_b @ M ) @ X )
        = X ) ) ).

% null_sets.Int_space_eq1
thf(fact_1037_space__restrict__space2,axiom,
    ! [Omega: set_b,M: sigma_measure_b] :
      ( ( member_set_b @ Omega @ ( sigma_sets_b @ M ) )
     => ( ( sigma_space_b @ ( sigma_8692839461743104067pace_b @ M @ Omega ) )
        = Omega ) ) ).

% space_restrict_space2
thf(fact_1038_space__restrict__space2,axiom,
    ! [Omega: set_d,M: sigma_measure_d] :
      ( ( member_set_d @ Omega @ ( sigma_sets_d @ M ) )
     => ( ( sigma_space_d @ ( sigma_8692839461743104069pace_d @ M @ Omega ) )
        = Omega ) ) ).

% space_restrict_space2
thf(fact_1039_sets_Ospace__measure__of__eq,axiom,
    ! [M: sigma_measure_b,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( sigma_space_b @ ( sigma_measure_of_b @ ( sigma_space_b @ M ) @ ( sigma_sets_b @ M ) @ Mu2 ) )
      = ( sigma_space_b @ M ) ) ).

% sets.space_measure_of_eq
thf(fact_1040_sets_Ospace__measure__of__eq,axiom,
    ! [M: sigma_measure_d,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( sigma_space_d @ ( sigma_measure_of_d @ ( sigma_space_d @ M ) @ ( sigma_sets_d @ M ) @ Mu2 ) )
      = ( sigma_space_d @ M ) ) ).

% sets.space_measure_of_eq
thf(fact_1041_sets_Osets__measure__of__eq,axiom,
    ! [M: sigma_measure_b,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( sigma_sets_b @ ( sigma_measure_of_b @ ( sigma_space_b @ M ) @ ( sigma_sets_b @ M ) @ Mu2 ) )
      = ( sigma_sets_b @ M ) ) ).

% sets.sets_measure_of_eq
thf(fact_1042_sets_Osets__measure__of__eq,axiom,
    ! [M: sigma_measure_d,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( sigma_sets_d @ ( sigma_measure_of_d @ ( sigma_space_d @ M ) @ ( sigma_sets_d @ M ) @ Mu2 ) )
      = ( sigma_sets_d @ M ) ) ).

% sets.sets_measure_of_eq
thf(fact_1043_in__measure__of,axiom,
    ! [M: set_set_d,Omega: set_d,A: set_d,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ M @ ( pow_d @ Omega ) )
     => ( ( member_set_d @ A @ M )
       => ( member_set_d @ A @ ( sigma_sets_d @ ( sigma_measure_of_d @ Omega @ M @ Mu2 ) ) ) ) ) ).

% in_measure_of
thf(fact_1044_in__measure__of,axiom,
    ! [M: set_set_b,Omega: set_b,A: set_b,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ M @ ( pow_b @ Omega ) )
     => ( ( member_set_b @ A @ M )
       => ( member_set_b @ A @ ( sigma_sets_b @ ( sigma_measure_of_b @ Omega @ M @ Mu2 ) ) ) ) ) ).

% in_measure_of
thf(fact_1045_space__measure__of,axiom,
    ! [A: set_set_d,Omega: set_d,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ A @ ( pow_d @ Omega ) )
     => ( ( sigma_space_d @ ( sigma_measure_of_d @ Omega @ A @ Mu2 ) )
        = Omega ) ) ).

% space_measure_of
thf(fact_1046_space__measure__of,axiom,
    ! [A: set_set_b,Omega: set_b,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( pow_b @ Omega ) )
     => ( ( sigma_space_b @ ( sigma_measure_of_b @ Omega @ A @ Mu2 ) )
        = Omega ) ) ).

% space_measure_of
thf(fact_1047_sets__measure__of,axiom,
    ! [A: set_set_d,Omega: set_d,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ A @ ( pow_d @ Omega ) )
     => ( ( sigma_sets_d @ ( sigma_measure_of_d @ Omega @ A @ Mu2 ) )
        = ( sigma_sigma_sets_d @ Omega @ A ) ) ) ).

% sets_measure_of
thf(fact_1048_sets__measure__of,axiom,
    ! [A: set_set_b,Omega: set_b,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( pow_b @ Omega ) )
     => ( ( sigma_sets_b @ ( sigma_measure_of_b @ Omega @ A @ Mu2 ) )
        = ( sigma_sigma_sets_b @ Omega @ A ) ) ) ).

% sets_measure_of
thf(fact_1049_space__in__measure__of,axiom,
    ! [Omega: set_d,M: set_set_d,Mu2: set_d > extend8495563244428889912nnreal] : ( member_set_d @ Omega @ ( sigma_sets_d @ ( sigma_measure_of_d @ Omega @ M @ Mu2 ) ) ) ).

% space_in_measure_of
thf(fact_1050_space__in__measure__of,axiom,
    ! [Omega: set_b,M: set_set_b,Mu2: set_b > extend8495563244428889912nnreal] : ( member_set_b @ Omega @ ( sigma_sets_b @ ( sigma_measure_of_b @ Omega @ M @ Mu2 ) ) ) ).

% space_in_measure_of
thf(fact_1051_space__measure__of__conv,axiom,
    ! [Omega: set_b,A: set_set_b,Mu2: set_b > extend8495563244428889912nnreal] :
      ( ( sigma_space_b @ ( sigma_measure_of_b @ Omega @ A @ Mu2 ) )
      = Omega ) ).

% space_measure_of_conv
thf(fact_1052_space__measure__of__conv,axiom,
    ! [Omega: set_d,A: set_set_d,Mu2: set_d > extend8495563244428889912nnreal] :
      ( ( sigma_space_d @ ( sigma_measure_of_d @ Omega @ A @ Mu2 ) )
      = Omega ) ).

% space_measure_of_conv
thf(fact_1053_sets__restrict__space__cong,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d,Omega: set_d] :
      ( ( ( sigma_sets_d @ M )
        = ( sigma_sets_d @ N ) )
     => ( ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ M @ Omega ) )
        = ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ N @ Omega ) ) ) ) ).

% sets_restrict_space_cong
thf(fact_1054_sets__restrict__space__cong,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b,Omega: set_b] :
      ( ( ( sigma_sets_b @ M )
        = ( sigma_sets_b @ N ) )
     => ( ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ M @ Omega ) )
        = ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ N @ Omega ) ) ) ) ).

% sets_restrict_space_cong
thf(fact_1055_restrict__space__sets__cong,axiom,
    ! [A: set_d,B: set_d,M: sigma_measure_d,N: sigma_measure_d] :
      ( ( A = B )
     => ( ( ( sigma_sets_d @ M )
          = ( sigma_sets_d @ N ) )
       => ( ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ M @ A ) )
          = ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ N @ B ) ) ) ) ) ).

% restrict_space_sets_cong
thf(fact_1056_restrict__space__sets__cong,axiom,
    ! [A: set_b,B: set_b,M: sigma_measure_b,N: sigma_measure_b] :
      ( ( A = B )
     => ( ( ( sigma_sets_b @ M )
          = ( sigma_sets_b @ N ) )
       => ( ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ M @ A ) )
          = ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ N @ B ) ) ) ) ) ).

% restrict_space_sets_cong
thf(fact_1057_measurable__restrict__space1,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d,Omega: set_b] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
     => ( member_b_d @ F @ ( sigma_measurable_b_d @ ( sigma_8692839461743104067pace_b @ M @ Omega ) @ N ) ) ) ).

% measurable_restrict_space1
thf(fact_1058_measurable__restrict__space1,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,Omega: set_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( member_b_b @ F @ ( sigma_measurable_b_b @ ( sigma_8692839461743104067pace_b @ M @ Omega ) @ N ) ) ) ).

% measurable_restrict_space1
thf(fact_1059_measurable__restrict__space1,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b,Omega: set_d] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
     => ( member_d_b @ F @ ( sigma_measurable_d_b @ ( sigma_8692839461743104069pace_d @ M @ Omega ) @ N ) ) ) ).

% measurable_restrict_space1
thf(fact_1060_measurable__restrict__space1,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d,Omega: set_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
     => ( member_d_d @ F @ ( sigma_measurable_d_d @ ( sigma_8692839461743104069pace_d @ M @ Omega ) @ N ) ) ) ).

% measurable_restrict_space1
thf(fact_1061_null__setsD2,axiom,
    ! [A: set_d,M: sigma_measure_d] :
      ( ( member_set_d @ A @ ( measure_null_sets_d @ M ) )
     => ( member_set_d @ A @ ( sigma_sets_d @ M ) ) ) ).

% null_setsD2
thf(fact_1062_null__setsD2,axiom,
    ! [A: set_b,M: sigma_measure_b] :
      ( ( member_set_b @ A @ ( measure_null_sets_b @ M ) )
     => ( member_set_b @ A @ ( sigma_sets_b @ M ) ) ) ).

% null_setsD2
thf(fact_1063_null__sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_set_d,P: set_d > $o,Q: set_d > $o] :
      ( ( member_set_set_d
        @ ( collect_set_d
          @ ^ [X2: set_d] :
              ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur1516554140942087188_set_d @ M ) )
     => ( ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur1516554140942087188_set_d @ M ) )
       => ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( measur1516554140942087188_set_d @ M ) ) ) ) ).

% null_sets.sets_Collect_disj
thf(fact_1064_null__sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_c,P: c > $o,Q: c > $o] :
      ( ( member_set_c
        @ ( collect_c
          @ ^ [X2: c] :
              ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
              & ( P @ X2 ) ) )
        @ ( measure_null_sets_c @ M ) )
     => ( ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measure_null_sets_c @ M ) )
       => ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( measure_null_sets_c @ M ) ) ) ) ).

% null_sets.sets_Collect_disj
thf(fact_1065_null__sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_b_d,P: ( b > d ) > $o,Q: ( b > d ) > $o] :
      ( ( member_set_b_d2
        @ ( collect_b_d
          @ ^ [X2: b > d] :
              ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur1875269531988832564ts_b_d @ M ) )
     => ( ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur1875269531988832564ts_b_d @ M ) )
       => ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( measur1875269531988832564ts_b_d @ M ) ) ) ) ).

% null_sets.sets_Collect_disj
thf(fact_1066_null__sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_b_b,P: ( b > b ) > $o,Q: ( b > b ) > $o] :
      ( ( member_set_b_b2
        @ ( collect_b_b
          @ ^ [X2: b > b] :
              ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur1875269523382374962ts_b_b @ M ) )
     => ( ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur1875269523382374962ts_b_b @ M ) )
       => ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( measur1875269523382374962ts_b_b @ M ) ) ) ) ).

% null_sets.sets_Collect_disj
thf(fact_1067_null__sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_d_b,P: ( d > b ) > $o,Q: ( d > b ) > $o] :
      ( ( member_set_d_b
        @ ( collect_d_b
          @ ^ [X2: d > b] :
              ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur5523730396291798704ts_d_b @ M ) )
     => ( ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur5523730396291798704ts_d_b @ M ) )
       => ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( measur5523730396291798704ts_d_b @ M ) ) ) ) ).

% null_sets.sets_Collect_disj
thf(fact_1068_null__sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_d_d,P: ( d > d ) > $o,Q: ( d > d ) > $o] :
      ( ( member_set_d_d
        @ ( collect_d_d
          @ ^ [X2: d > d] :
              ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur5523730404898256306ts_d_d @ M ) )
     => ( ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur5523730404898256306ts_d_d @ M ) )
       => ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( measur5523730404898256306ts_d_d @ M ) ) ) ) ).

% null_sets.sets_Collect_disj
thf(fact_1069_null__sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_set_b,P: set_b > $o,Q: set_b > $o] :
      ( ( member_set_set_b
        @ ( collect_set_b
          @ ^ [X2: set_b] :
              ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur1516554132335629586_set_b @ M ) )
     => ( ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur1516554132335629586_set_b @ M ) )
       => ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( measur1516554132335629586_set_b @ M ) ) ) ) ).

% null_sets.sets_Collect_disj
thf(fact_1070_null__sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( measure_null_sets_b @ M ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measure_null_sets_b @ M ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( measure_null_sets_b @ M ) ) ) ) ).

% null_sets.sets_Collect_disj
thf(fact_1071_null__sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_d,P: d > $o,Q: d > $o] :
      ( ( member_set_d
        @ ( collect_d
          @ ^ [X2: d] :
              ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( measure_null_sets_d @ M ) )
     => ( ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measure_null_sets_d @ M ) )
       => ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( ( Q @ X2 )
                  | ( P @ X2 ) ) ) )
          @ ( measure_null_sets_d @ M ) ) ) ) ).

% null_sets.sets_Collect_disj
thf(fact_1072_null__sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_set_d,P: set_d > $o,Q: set_d > $o] :
      ( ( member_set_set_d
        @ ( collect_set_d
          @ ^ [X2: set_d] :
              ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur1516554140942087188_set_d @ M ) )
     => ( ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur1516554140942087188_set_d @ M ) )
       => ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( measur1516554140942087188_set_d @ M ) ) ) ) ).

% null_sets.sets_Collect_conj
thf(fact_1073_null__sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_c,P: c > $o,Q: c > $o] :
      ( ( member_set_c
        @ ( collect_c
          @ ^ [X2: c] :
              ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
              & ( P @ X2 ) ) )
        @ ( measure_null_sets_c @ M ) )
     => ( ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measure_null_sets_c @ M ) )
       => ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( measure_null_sets_c @ M ) ) ) ) ).

% null_sets.sets_Collect_conj
thf(fact_1074_null__sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_b_d,P: ( b > d ) > $o,Q: ( b > d ) > $o] :
      ( ( member_set_b_d2
        @ ( collect_b_d
          @ ^ [X2: b > d] :
              ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur1875269531988832564ts_b_d @ M ) )
     => ( ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur1875269531988832564ts_b_d @ M ) )
       => ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( measur1875269531988832564ts_b_d @ M ) ) ) ) ).

% null_sets.sets_Collect_conj
thf(fact_1075_null__sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_b_b,P: ( b > b ) > $o,Q: ( b > b ) > $o] :
      ( ( member_set_b_b2
        @ ( collect_b_b
          @ ^ [X2: b > b] :
              ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur1875269523382374962ts_b_b @ M ) )
     => ( ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur1875269523382374962ts_b_b @ M ) )
       => ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( measur1875269523382374962ts_b_b @ M ) ) ) ) ).

% null_sets.sets_Collect_conj
thf(fact_1076_null__sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_d_b,P: ( d > b ) > $o,Q: ( d > b ) > $o] :
      ( ( member_set_d_b
        @ ( collect_d_b
          @ ^ [X2: d > b] :
              ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur5523730396291798704ts_d_b @ M ) )
     => ( ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur5523730396291798704ts_d_b @ M ) )
       => ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( measur5523730396291798704ts_d_b @ M ) ) ) ) ).

% null_sets.sets_Collect_conj
thf(fact_1077_null__sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_d_d,P: ( d > d ) > $o,Q: ( d > d ) > $o] :
      ( ( member_set_d_d
        @ ( collect_d_d
          @ ^ [X2: d > d] :
              ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur5523730404898256306ts_d_d @ M ) )
     => ( ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur5523730404898256306ts_d_d @ M ) )
       => ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( measur5523730404898256306ts_d_d @ M ) ) ) ) ).

% null_sets.sets_Collect_conj
thf(fact_1078_null__sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_set_b,P: set_b > $o,Q: set_b > $o] :
      ( ( member_set_set_b
        @ ( collect_set_b
          @ ^ [X2: set_b] :
              ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( measur1516554132335629586_set_b @ M ) )
     => ( ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measur1516554132335629586_set_b @ M ) )
       => ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( measur1516554132335629586_set_b @ M ) ) ) ) ).

% null_sets.sets_Collect_conj
thf(fact_1079_null__sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( measure_null_sets_b @ M ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measure_null_sets_b @ M ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( measure_null_sets_b @ M ) ) ) ) ).

% null_sets.sets_Collect_conj
thf(fact_1080_null__sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_d,P: d > $o,Q: d > $o] :
      ( ( member_set_d
        @ ( collect_d
          @ ^ [X2: d] :
              ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( measure_null_sets_d @ M ) )
     => ( ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( Q @ X2 ) ) )
          @ ( measure_null_sets_d @ M ) )
       => ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( Q @ X2 )
                & ( P @ X2 ) ) )
          @ ( measure_null_sets_d @ M ) ) ) ) ).

% null_sets.sets_Collect_conj
thf(fact_1081_measure__restrict__space,axiom,
    ! [Omega: set_d,M: sigma_measure_d,A: set_d] :
      ( ( member_set_d @ ( inf_inf_set_d @ Omega @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) )
     => ( ( ord_less_eq_set_d @ A @ Omega )
       => ( ( sigma_measure_d2 @ ( sigma_8692839461743104069pace_d @ M @ Omega ) @ A )
          = ( sigma_measure_d2 @ M @ A ) ) ) ) ).

% measure_restrict_space
thf(fact_1082_measure__restrict__space,axiom,
    ! [Omega: set_set_b,M: sigma_measure_set_b,A: set_set_b] :
      ( ( member_set_set_b @ ( inf_inf_set_set_b @ Omega @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) )
     => ( ( ord_le3795704787696855135_set_b @ A @ Omega )
       => ( ( sigma_measure_set_b2 @ ( sigma_5799942775684512035_set_b @ M @ Omega ) @ A )
          = ( sigma_measure_set_b2 @ M @ A ) ) ) ) ).

% measure_restrict_space
thf(fact_1083_measure__restrict__space,axiom,
    ! [Omega: set_b,M: sigma_measure_b,A: set_b] :
      ( ( member_set_b @ ( inf_inf_set_b @ Omega @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
     => ( ( ord_less_eq_set_b @ A @ Omega )
       => ( ( sigma_measure_b2 @ ( sigma_8692839461743104067pace_b @ M @ Omega ) @ A )
          = ( sigma_measure_b2 @ M @ A ) ) ) ) ).

% measure_restrict_space
thf(fact_1084_sets__restrict__restrict__space,axiom,
    ! [M: sigma_measure_b,A: set_b,B: set_b] :
      ( ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ ( sigma_8692839461743104067pace_b @ M @ A ) @ B ) )
      = ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ M @ ( inf_inf_set_b @ A @ B ) ) ) ) ).

% sets_restrict_restrict_space
thf(fact_1085_sets__restrict__restrict__space,axiom,
    ! [M: sigma_measure_d,A: set_d,B: set_d] :
      ( ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ ( sigma_8692839461743104069pace_d @ M @ A ) @ B ) )
      = ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ M @ ( inf_inf_set_d @ A @ B ) ) ) ) ).

% sets_restrict_restrict_space
thf(fact_1086_sets__restrict__restrict__space,axiom,
    ! [M: sigma_measure_set_b,A: set_set_b,B: set_set_b] :
      ( ( sigma_sets_set_b @ ( sigma_5799942775684512035_set_b @ ( sigma_5799942775684512035_set_b @ M @ A ) @ B ) )
      = ( sigma_sets_set_b @ ( sigma_5799942775684512035_set_b @ M @ ( inf_inf_set_set_b @ A @ B ) ) ) ) ).

% sets_restrict_restrict_space
thf(fact_1087_space__restrict__space,axiom,
    ! [M: sigma_measure_b,Omega: set_b] :
      ( ( sigma_space_b @ ( sigma_8692839461743104067pace_b @ M @ Omega ) )
      = ( inf_inf_set_b @ Omega @ ( sigma_space_b @ M ) ) ) ).

% space_restrict_space
thf(fact_1088_space__restrict__space,axiom,
    ! [M: sigma_measure_d,Omega: set_d] :
      ( ( sigma_space_d @ ( sigma_8692839461743104069pace_d @ M @ Omega ) )
      = ( inf_inf_set_d @ Omega @ ( sigma_space_d @ M ) ) ) ).

% space_restrict_space
thf(fact_1089_space__restrict__space,axiom,
    ! [M: sigma_measure_set_b,Omega: set_set_b] :
      ( ( sigma_space_set_b @ ( sigma_5799942775684512035_set_b @ M @ Omega ) )
      = ( inf_inf_set_set_b @ Omega @ ( sigma_space_set_b @ M ) ) ) ).

% space_restrict_space
thf(fact_1090_mono__restrict__space,axiom,
    ! [M: sigma_measure_d,N: sigma_measure_d,X4: set_d] :
      ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M ) @ ( sigma_sets_d @ N ) )
     => ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ M @ X4 ) ) @ ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ N @ X4 ) ) ) ) ).

% mono_restrict_space
thf(fact_1091_mono__restrict__space,axiom,
    ! [M: sigma_measure_b,N: sigma_measure_b,X4: set_b] :
      ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M ) @ ( sigma_sets_b @ N ) )
     => ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ M @ X4 ) ) @ ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ N @ X4 ) ) ) ) ).

% mono_restrict_space
thf(fact_1092_null__sets_Osets__into__space,axiom,
    ! [X: set_d,M: sigma_measure_d] :
      ( ( member_set_d @ X @ ( measure_null_sets_d @ M ) )
     => ( ord_less_eq_set_d @ X @ ( sigma_space_d @ M ) ) ) ).

% null_sets.sets_into_space
thf(fact_1093_null__sets_Osets__into__space,axiom,
    ! [X: set_set_b,M: sigma_measure_set_b] :
      ( ( member_set_set_b @ X @ ( measur1516554132335629586_set_b @ M ) )
     => ( ord_le3795704787696855135_set_b @ X @ ( sigma_space_set_b @ M ) ) ) ).

% null_sets.sets_into_space
thf(fact_1094_null__sets_Osets__into__space,axiom,
    ! [X: set_b,M: sigma_measure_b] :
      ( ( member_set_b @ X @ ( measure_null_sets_b @ M ) )
     => ( ord_less_eq_set_b @ X @ ( sigma_space_b @ M ) ) ) ).

% null_sets.sets_into_space
thf(fact_1095_null__set__Int2,axiom,
    ! [B: set_b,M: sigma_measure_b,A: set_b] :
      ( ( member_set_b @ B @ ( measure_null_sets_b @ M ) )
     => ( ( member_set_b @ A @ ( sigma_sets_b @ M ) )
       => ( member_set_b @ ( inf_inf_set_b @ B @ A ) @ ( measure_null_sets_b @ M ) ) ) ) ).

% null_set_Int2
thf(fact_1096_null__set__Int2,axiom,
    ! [B: set_d,M: sigma_measure_d,A: set_d] :
      ( ( member_set_d @ B @ ( measure_null_sets_d @ M ) )
     => ( ( member_set_d @ A @ ( sigma_sets_d @ M ) )
       => ( member_set_d @ ( inf_inf_set_d @ B @ A ) @ ( measure_null_sets_d @ M ) ) ) ) ).

% null_set_Int2
thf(fact_1097_null__set__Int2,axiom,
    ! [B: set_set_b,M: sigma_measure_set_b,A: set_set_b] :
      ( ( member_set_set_b @ B @ ( measur1516554132335629586_set_b @ M ) )
     => ( ( member_set_set_b @ A @ ( sigma_sets_set_b @ M ) )
       => ( member_set_set_b @ ( inf_inf_set_set_b @ B @ A ) @ ( measur1516554132335629586_set_b @ M ) ) ) ) ).

% null_set_Int2
thf(fact_1098_null__set__Int1,axiom,
    ! [B: set_b,M: sigma_measure_b,A: set_b] :
      ( ( member_set_b @ B @ ( measure_null_sets_b @ M ) )
     => ( ( member_set_b @ A @ ( sigma_sets_b @ M ) )
       => ( member_set_b @ ( inf_inf_set_b @ A @ B ) @ ( measure_null_sets_b @ M ) ) ) ) ).

% null_set_Int1
thf(fact_1099_null__set__Int1,axiom,
    ! [B: set_d,M: sigma_measure_d,A: set_d] :
      ( ( member_set_d @ B @ ( measure_null_sets_d @ M ) )
     => ( ( member_set_d @ A @ ( sigma_sets_d @ M ) )
       => ( member_set_d @ ( inf_inf_set_d @ A @ B ) @ ( measure_null_sets_d @ M ) ) ) ) ).

% null_set_Int1
thf(fact_1100_null__set__Int1,axiom,
    ! [B: set_set_b,M: sigma_measure_set_b,A: set_set_b] :
      ( ( member_set_set_b @ B @ ( measur1516554132335629586_set_b @ M ) )
     => ( ( member_set_set_b @ A @ ( sigma_sets_set_b @ M ) )
       => ( member_set_set_b @ ( inf_inf_set_set_b @ A @ B ) @ ( measur1516554132335629586_set_b @ M ) ) ) ) ).

% null_set_Int1
thf(fact_1101_sets__Collect__restrict__space__iff,axiom,
    ! [S: set_set_d,M: sigma_measure_set_d,P: set_d > $o] :
      ( ( member_set_set_d @ S @ ( sigma_sets_set_d @ M ) )
     => ( ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ ( sigma_5799942784290969637_set_d @ M @ S ) ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_set_d @ ( sigma_5799942784290969637_set_d @ M @ S ) ) )
        = ( member_set_set_d
          @ ( collect_set_d
            @ ^ [X2: set_d] :
                ( ( member_set_d @ X2 @ ( sigma_space_set_d @ M ) )
                & ( member_set_d @ X2 @ S )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_set_d @ M ) ) ) ) ).

% sets_Collect_restrict_space_iff
thf(fact_1102_sets__Collect__restrict__space__iff,axiom,
    ! [S: set_c,M: sigma_measure_c,P: c > $o] :
      ( ( member_set_c @ S @ ( sigma_sets_c @ M ) )
     => ( ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ ( sigma_8692839461743104068pace_c @ M @ S ) ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_c @ ( sigma_8692839461743104068pace_c @ M @ S ) ) )
        = ( member_set_c
          @ ( collect_c
            @ ^ [X2: c] :
                ( ( member_c @ X2 @ ( sigma_space_c @ M ) )
                & ( member_c @ X2 @ S )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_c @ M ) ) ) ) ).

% sets_Collect_restrict_space_iff
thf(fact_1103_sets__Collect__restrict__space__iff,axiom,
    ! [S: set_b_d,M: sigma_measure_b_d,P: ( b > d ) > $o] :
      ( ( member_set_b_d2 @ S @ ( sigma_sets_b_d @ M ) )
     => ( ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ ( sigma_547908261065455843ce_b_d @ M @ S ) ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b_d @ ( sigma_547908261065455843ce_b_d @ M @ S ) ) )
        = ( member_set_b_d2
          @ ( collect_b_d
            @ ^ [X2: b > d] :
                ( ( member_b_d @ X2 @ ( sigma_space_b_d @ M ) )
                & ( member_b_d @ X2 @ S )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b_d @ M ) ) ) ) ).

% sets_Collect_restrict_space_iff
thf(fact_1104_sets__Collect__restrict__space__iff,axiom,
    ! [S: set_b_b,M: sigma_measure_b_b,P: ( b > b ) > $o] :
      ( ( member_set_b_b2 @ S @ ( sigma_sets_b_b @ M ) )
     => ( ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ ( sigma_547908252458998241ce_b_b @ M @ S ) ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b_b @ ( sigma_547908252458998241ce_b_b @ M @ S ) ) )
        = ( member_set_b_b2
          @ ( collect_b_b
            @ ^ [X2: b > b] :
                ( ( member_b_b @ X2 @ ( sigma_space_b_b @ M ) )
                & ( member_b_b @ X2 @ S )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b_b @ M ) ) ) ) ).

% sets_Collect_restrict_space_iff
thf(fact_1105_sets__Collect__restrict__space__iff,axiom,
    ! [S: set_d_b,M: sigma_measure_d_b,P: ( d > b ) > $o] :
      ( ( member_set_d_b @ S @ ( sigma_sets_d_b @ M ) )
     => ( ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ ( sigma_4196369125368421983ce_d_b @ M @ S ) ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d_b @ ( sigma_4196369125368421983ce_d_b @ M @ S ) ) )
        = ( member_set_d_b
          @ ( collect_d_b
            @ ^ [X2: d > b] :
                ( ( member_d_b @ X2 @ ( sigma_space_d_b @ M ) )
                & ( member_d_b @ X2 @ S )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d_b @ M ) ) ) ) ).

% sets_Collect_restrict_space_iff
thf(fact_1106_sets__Collect__restrict__space__iff,axiom,
    ! [S: set_d_d,M: sigma_measure_d_d,P: ( d > d ) > $o] :
      ( ( member_set_d_d @ S @ ( sigma_sets_d_d @ M ) )
     => ( ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ ( sigma_4196369133974879585ce_d_d @ M @ S ) ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d_d @ ( sigma_4196369133974879585ce_d_d @ M @ S ) ) )
        = ( member_set_d_d
          @ ( collect_d_d
            @ ^ [X2: d > d] :
                ( ( member_d_d @ X2 @ ( sigma_space_d_d @ M ) )
                & ( member_d_d @ X2 @ S )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d_d @ M ) ) ) ) ).

% sets_Collect_restrict_space_iff
thf(fact_1107_sets__Collect__restrict__space__iff,axiom,
    ! [S: set_set_b,M: sigma_measure_set_b,P: set_b > $o] :
      ( ( member_set_set_b @ S @ ( sigma_sets_set_b @ M ) )
     => ( ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ ( sigma_5799942775684512035_set_b @ M @ S ) ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_set_b @ ( sigma_5799942775684512035_set_b @ M @ S ) ) )
        = ( member_set_set_b
          @ ( collect_set_b
            @ ^ [X2: set_b] :
                ( ( member_set_b @ X2 @ ( sigma_space_set_b @ M ) )
                & ( member_set_b @ X2 @ S )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_set_b @ M ) ) ) ) ).

% sets_Collect_restrict_space_iff
thf(fact_1108_sets__Collect__restrict__space__iff,axiom,
    ! [S: set_b,M: sigma_measure_b,P: b > $o] :
      ( ( member_set_b @ S @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ ( sigma_8692839461743104067pace_b @ M @ S ) ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ M @ S ) ) )
        = ( member_set_b
          @ ( collect_b
            @ ^ [X2: b] :
                ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
                & ( member_b @ X2 @ S )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% sets_Collect_restrict_space_iff
thf(fact_1109_sets__Collect__restrict__space__iff,axiom,
    ! [S: set_d,M: sigma_measure_d,P: d > $o] :
      ( ( member_set_d @ S @ ( sigma_sets_d @ M ) )
     => ( ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ ( sigma_8692839461743104069pace_d @ M @ S ) ) )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ M @ S ) ) )
        = ( member_set_d
          @ ( collect_d
            @ ^ [X2: d] :
                ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
                & ( member_d @ X2 @ S )
                & ( P @ X2 ) ) )
          @ ( sigma_sets_d @ M ) ) ) ) ).

% sets_Collect_restrict_space_iff
thf(fact_1110_measure__of__of__measure,axiom,
    ! [M: sigma_measure_b] :
      ( ( sigma_measure_of_b @ ( sigma_space_b @ M ) @ ( sigma_sets_b @ M ) @ ( sigma_emeasure_b @ M ) )
      = M ) ).

% measure_of_of_measure
thf(fact_1111_measure__of__of__measure,axiom,
    ! [M: sigma_measure_d] :
      ( ( sigma_measure_of_d @ ( sigma_space_d @ M ) @ ( sigma_sets_d @ M ) @ ( sigma_emeasure_d @ M ) )
      = M ) ).

% measure_of_of_measure
thf(fact_1112_measure__of__subset,axiom,
    ! [M: set_set_d,Omega: set_d,M2: set_set_d,Mu2: set_d > extend8495563244428889912nnreal,Mu3: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ M @ ( pow_d @ Omega ) )
     => ( ( ord_le3937772867789547105_set_d @ M2 @ M )
       => ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ ( sigma_measure_of_d @ Omega @ M2 @ Mu2 ) ) @ ( sigma_sets_d @ ( sigma_measure_of_d @ Omega @ M @ Mu3 ) ) ) ) ) ).

% measure_of_subset
thf(fact_1113_measure__of__subset,axiom,
    ! [M: set_set_b,Omega: set_b,M2: set_set_b,Mu2: set_b > extend8495563244428889912nnreal,Mu3: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ M @ ( pow_b @ Omega ) )
     => ( ( ord_le3795704787696855135_set_b @ M2 @ M )
       => ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ ( sigma_measure_of_b @ Omega @ M2 @ Mu2 ) ) @ ( sigma_sets_b @ ( sigma_measure_of_b @ Omega @ M @ Mu3 ) ) ) ) ) ).

% measure_of_subset
thf(fact_1114_measure__of__eq,axiom,
    ! [A: set_set_d,Omega: set_d,Mu2: set_d > extend8495563244428889912nnreal,Mu3: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ A @ ( pow_d @ Omega ) )
     => ( ! [A6: set_d] :
            ( ( member_set_d @ A6 @ ( sigma_sigma_sets_d @ Omega @ A ) )
           => ( ( Mu2 @ A6 )
              = ( Mu3 @ A6 ) ) )
       => ( ( sigma_measure_of_d @ Omega @ A @ Mu2 )
          = ( sigma_measure_of_d @ Omega @ A @ Mu3 ) ) ) ) ).

% measure_of_eq
thf(fact_1115_measure__of__eq,axiom,
    ! [A: set_set_b,Omega: set_b,Mu2: set_b > extend8495563244428889912nnreal,Mu3: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( pow_b @ Omega ) )
     => ( ! [A6: set_b] :
            ( ( member_set_b @ A6 @ ( sigma_sigma_sets_b @ Omega @ A ) )
           => ( ( Mu2 @ A6 )
              = ( Mu3 @ A6 ) ) )
       => ( ( sigma_measure_of_b @ Omega @ A @ Mu2 )
          = ( sigma_measure_of_b @ Omega @ A @ Mu3 ) ) ) ) ).

% measure_of_eq
thf(fact_1116_restrict__restrict__space,axiom,
    ! [A: set_b,M: sigma_measure_b,B: set_b] :
      ( ( member_set_b @ ( inf_inf_set_b @ A @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b @ ( inf_inf_set_b @ B @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
       => ( ( sigma_8692839461743104067pace_b @ ( sigma_8692839461743104067pace_b @ M @ A ) @ B )
          = ( sigma_8692839461743104067pace_b @ M @ ( inf_inf_set_b @ A @ B ) ) ) ) ) ).

% restrict_restrict_space
thf(fact_1117_restrict__restrict__space,axiom,
    ! [A: set_d,M: sigma_measure_d,B: set_d] :
      ( ( member_set_d @ ( inf_inf_set_d @ A @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) )
     => ( ( member_set_d @ ( inf_inf_set_d @ B @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) )
       => ( ( sigma_8692839461743104069pace_d @ ( sigma_8692839461743104069pace_d @ M @ A ) @ B )
          = ( sigma_8692839461743104069pace_d @ M @ ( inf_inf_set_d @ A @ B ) ) ) ) ) ).

% restrict_restrict_space
thf(fact_1118_restrict__restrict__space,axiom,
    ! [A: set_set_b,M: sigma_measure_set_b,B: set_set_b] :
      ( ( member_set_set_b @ ( inf_inf_set_set_b @ A @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) )
     => ( ( member_set_set_b @ ( inf_inf_set_set_b @ B @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) )
       => ( ( sigma_5799942775684512035_set_b @ ( sigma_5799942775684512035_set_b @ M @ A ) @ B )
          = ( sigma_5799942775684512035_set_b @ M @ ( inf_inf_set_set_b @ A @ B ) ) ) ) ) ).

% restrict_restrict_space
thf(fact_1119_null__sets_Ospace__closed,axiom,
    ! [M: sigma_measure_d] : ( ord_le3937772867789547105_set_d @ ( measure_null_sets_d @ M ) @ ( pow_d @ ( sigma_space_d @ M ) ) ) ).

% null_sets.space_closed
thf(fact_1120_null__sets_Ospace__closed,axiom,
    ! [M: sigma_measure_b] : ( ord_le3795704787696855135_set_b @ ( measure_null_sets_b @ M ) @ ( pow_b @ ( sigma_space_b @ M ) ) ) ).

% null_sets.space_closed
thf(fact_1121_measurable__If__restrict__space__iff,axiom,
    ! [M: sigma_measure_b,P: b > $o,F: b > d,G: b > d,N: sigma_measure_d] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b @ M ) )
     => ( ( member_b_d
          @ ^ [X2: b] : ( if_d @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
          @ ( sigma_measurable_b_d @ M @ N ) )
        = ( ( member_b_d @ F @ ( sigma_measurable_b_d @ ( sigma_8692839461743104067pace_b @ M @ ( collect_b @ P ) ) @ N ) )
          & ( member_b_d @ G
            @ ( sigma_measurable_b_d
              @ ( sigma_8692839461743104067pace_b @ M
                @ ( collect_b
                  @ ^ [X2: b] :
                      ~ ( P @ X2 ) ) )
              @ N ) ) ) ) ) ).

% measurable_If_restrict_space_iff
thf(fact_1122_measurable__If__restrict__space__iff,axiom,
    ! [M: sigma_measure_b,P: b > $o,F: b > b,G: b > b,N: sigma_measure_b] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X2: b] :
              ( ( member_b @ X2 @ ( sigma_space_b @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_b @ M ) )
     => ( ( member_b_b
          @ ^ [X2: b] : ( if_b @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
          @ ( sigma_measurable_b_b @ M @ N ) )
        = ( ( member_b_b @ F @ ( sigma_measurable_b_b @ ( sigma_8692839461743104067pace_b @ M @ ( collect_b @ P ) ) @ N ) )
          & ( member_b_b @ G
            @ ( sigma_measurable_b_b
              @ ( sigma_8692839461743104067pace_b @ M
                @ ( collect_b
                  @ ^ [X2: b] :
                      ~ ( P @ X2 ) ) )
              @ N ) ) ) ) ) ).

% measurable_If_restrict_space_iff
thf(fact_1123_measurable__If__restrict__space__iff,axiom,
    ! [M: sigma_measure_d,P: d > $o,F: d > b,G: d > b,N: sigma_measure_b] :
      ( ( member_set_d
        @ ( collect_d
          @ ^ [X2: d] :
              ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d @ M ) )
     => ( ( member_d_b
          @ ^ [X2: d] : ( if_b @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
          @ ( sigma_measurable_d_b @ M @ N ) )
        = ( ( member_d_b @ F @ ( sigma_measurable_d_b @ ( sigma_8692839461743104069pace_d @ M @ ( collect_d @ P ) ) @ N ) )
          & ( member_d_b @ G
            @ ( sigma_measurable_d_b
              @ ( sigma_8692839461743104069pace_d @ M
                @ ( collect_d
                  @ ^ [X2: d] :
                      ~ ( P @ X2 ) ) )
              @ N ) ) ) ) ) ).

% measurable_If_restrict_space_iff
thf(fact_1124_measurable__If__restrict__space__iff,axiom,
    ! [M: sigma_measure_d,P: d > $o,F: d > d,G: d > d,N: sigma_measure_d] :
      ( ( member_set_d
        @ ( collect_d
          @ ^ [X2: d] :
              ( ( member_d @ X2 @ ( sigma_space_d @ M ) )
              & ( P @ X2 ) ) )
        @ ( sigma_sets_d @ M ) )
     => ( ( member_d_d
          @ ^ [X2: d] : ( if_d @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
          @ ( sigma_measurable_d_d @ M @ N ) )
        = ( ( member_d_d @ F @ ( sigma_measurable_d_d @ ( sigma_8692839461743104069pace_d @ M @ ( collect_d @ P ) ) @ N ) )
          & ( member_d_d @ G
            @ ( sigma_measurable_d_d
              @ ( sigma_8692839461743104069pace_d @ M
                @ ( collect_d
                  @ ^ [X2: d] :
                      ~ ( P @ X2 ) ) )
              @ N ) ) ) ) ) ).

% measurable_If_restrict_space_iff
thf(fact_1125_measurable__restrict__space2,axiom,
    ! [F: b > d,M: sigma_measure_b,Omega: set_d,N: sigma_measure_d] :
      ( ( member_b_d @ F
        @ ( pi_b_d @ ( sigma_space_b @ M )
          @ ^ [Uu: b] : Omega ) )
     => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
       => ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ ( sigma_8692839461743104069pace_d @ N @ Omega ) ) ) ) ) ).

% measurable_restrict_space2
thf(fact_1126_measurable__restrict__space2,axiom,
    ! [F: b > b,M: sigma_measure_b,Omega: set_b,N: sigma_measure_b] :
      ( ( member_b_b @ F
        @ ( pi_b_b @ ( sigma_space_b @ M )
          @ ^ [Uu: b] : Omega ) )
     => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
       => ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ ( sigma_8692839461743104067pace_b @ N @ Omega ) ) ) ) ) ).

% measurable_restrict_space2
thf(fact_1127_measurable__restrict__space2,axiom,
    ! [F: d > b,M: sigma_measure_d,Omega: set_b,N: sigma_measure_b] :
      ( ( member_d_b @ F
        @ ( pi_d_b @ ( sigma_space_d @ M )
          @ ^ [Uu: d] : Omega ) )
     => ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
       => ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ ( sigma_8692839461743104067pace_b @ N @ Omega ) ) ) ) ) ).

% measurable_restrict_space2
thf(fact_1128_measurable__restrict__space2,axiom,
    ! [F: d > d,M: sigma_measure_d,Omega: set_d,N: sigma_measure_d] :
      ( ( member_d_d @ F
        @ ( pi_d_d @ ( sigma_space_d @ M )
          @ ^ [Uu: d] : Omega ) )
     => ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
       => ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ ( sigma_8692839461743104069pace_d @ N @ Omega ) ) ) ) ) ).

% measurable_restrict_space2
thf(fact_1129_measurable__restrict__space2__iff,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d,Omega: set_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ ( sigma_8692839461743104069pace_d @ N @ Omega ) ) )
      = ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
        & ( member_b_d @ F
          @ ( pi_b_d @ ( sigma_space_b @ M )
            @ ^ [Uu: b] : Omega ) ) ) ) ).

% measurable_restrict_space2_iff
thf(fact_1130_measurable__restrict__space2__iff,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,Omega: set_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ ( sigma_8692839461743104067pace_b @ N @ Omega ) ) )
      = ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
        & ( member_b_b @ F
          @ ( pi_b_b @ ( sigma_space_b @ M )
            @ ^ [Uu: b] : Omega ) ) ) ) ).

% measurable_restrict_space2_iff
thf(fact_1131_measurable__restrict__space2__iff,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b,Omega: set_b] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ ( sigma_8692839461743104067pace_b @ N @ Omega ) ) )
      = ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
        & ( member_d_b @ F
          @ ( pi_d_b @ ( sigma_space_d @ M )
            @ ^ [Uu: d] : Omega ) ) ) ) ).

% measurable_restrict_space2_iff
thf(fact_1132_measurable__restrict__space2__iff,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d,Omega: set_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ ( sigma_8692839461743104069pace_d @ N @ Omega ) ) )
      = ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
        & ( member_d_d @ F
          @ ( pi_d_d @ ( sigma_space_d @ M )
            @ ^ [Uu: d] : Omega ) ) ) ) ).

% measurable_restrict_space2_iff
thf(fact_1133_sets__restrict__space__iff,axiom,
    ! [Omega: set_d,M: sigma_measure_d,A: set_d] :
      ( ( member_set_d @ ( inf_inf_set_d @ Omega @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) )
     => ( ( member_set_d @ A @ ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ M @ Omega ) ) )
        = ( ( ord_less_eq_set_d @ A @ Omega )
          & ( member_set_d @ A @ ( sigma_sets_d @ M ) ) ) ) ) ).

% sets_restrict_space_iff
thf(fact_1134_sets__restrict__space__iff,axiom,
    ! [Omega: set_set_b,M: sigma_measure_set_b,A: set_set_b] :
      ( ( member_set_set_b @ ( inf_inf_set_set_b @ Omega @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) )
     => ( ( member_set_set_b @ A @ ( sigma_sets_set_b @ ( sigma_5799942775684512035_set_b @ M @ Omega ) ) )
        = ( ( ord_le3795704787696855135_set_b @ A @ Omega )
          & ( member_set_set_b @ A @ ( sigma_sets_set_b @ M ) ) ) ) ) ).

% sets_restrict_space_iff
thf(fact_1135_sets__restrict__space__iff,axiom,
    ! [Omega: set_b,M: sigma_measure_b,A: set_b] :
      ( ( member_set_b @ ( inf_inf_set_b @ Omega @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b @ A @ ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ M @ Omega ) ) )
        = ( ( ord_less_eq_set_b @ A @ Omega )
          & ( member_set_b @ A @ ( sigma_sets_b @ M ) ) ) ) ) ).

% sets_restrict_space_iff
thf(fact_1136_measurable__mono1,axiom,
    ! [M2: set_set_d,Omega: set_d,M: set_set_d,Mu2: set_d > extend8495563244428889912nnreal,N: sigma_measure_b,Mu3: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ M2 @ ( pow_d @ Omega ) )
     => ( ( ord_le3937772867789547105_set_d @ M @ M2 )
       => ( ord_less_eq_set_d_b @ ( sigma_measurable_d_b @ ( sigma_measure_of_d @ Omega @ M @ Mu2 ) @ N ) @ ( sigma_measurable_d_b @ ( sigma_measure_of_d @ Omega @ M2 @ Mu3 ) @ N ) ) ) ) ).

% measurable_mono1
thf(fact_1137_measurable__mono1,axiom,
    ! [M2: set_set_d,Omega: set_d,M: set_set_d,Mu2: set_d > extend8495563244428889912nnreal,N: sigma_measure_d,Mu3: set_d > extend8495563244428889912nnreal] :
      ( ( ord_le3937772867789547105_set_d @ M2 @ ( pow_d @ Omega ) )
     => ( ( ord_le3937772867789547105_set_d @ M @ M2 )
       => ( ord_less_eq_set_d_d @ ( sigma_measurable_d_d @ ( sigma_measure_of_d @ Omega @ M @ Mu2 ) @ N ) @ ( sigma_measurable_d_d @ ( sigma_measure_of_d @ Omega @ M2 @ Mu3 ) @ N ) ) ) ) ).

% measurable_mono1
thf(fact_1138_measurable__mono1,axiom,
    ! [M2: set_set_b,Omega: set_b,M: set_set_b,Mu2: set_b > extend8495563244428889912nnreal,N: sigma_measure_d,Mu3: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ M2 @ ( pow_b @ Omega ) )
     => ( ( ord_le3795704787696855135_set_b @ M @ M2 )
       => ( ord_less_eq_set_b_d @ ( sigma_measurable_b_d @ ( sigma_measure_of_b @ Omega @ M @ Mu2 ) @ N ) @ ( sigma_measurable_b_d @ ( sigma_measure_of_b @ Omega @ M2 @ Mu3 ) @ N ) ) ) ) ).

% measurable_mono1
thf(fact_1139_measurable__mono1,axiom,
    ! [M2: set_set_b,Omega: set_b,M: set_set_b,Mu2: set_b > extend8495563244428889912nnreal,N: sigma_measure_b,Mu3: set_b > extend8495563244428889912nnreal] :
      ( ( ord_le3795704787696855135_set_b @ M2 @ ( pow_b @ Omega ) )
     => ( ( ord_le3795704787696855135_set_b @ M @ M2 )
       => ( ord_less_eq_set_b_b @ ( sigma_measurable_b_b @ ( sigma_measure_of_b @ Omega @ M @ Mu2 ) @ N ) @ ( sigma_measurable_b_b @ ( sigma_measure_of_b @ Omega @ M2 @ Mu3 ) @ N ) ) ) ) ).

% measurable_mono1
thf(fact_1140_measurable__restrict__space__iff,axiom,
    ! [Omega: set_c,M: sigma_measure_c,C2: c,N: sigma_measure_c,F: c > c] :
      ( ( member_set_c @ ( inf_inf_set_c @ Omega @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) )
     => ( ( member_c @ C2 @ ( sigma_space_c @ N ) )
       => ( ( member_c_c @ F @ ( sigma_measurable_c_c @ ( sigma_8692839461743104068pace_c @ M @ Omega ) @ N ) )
          = ( member_c_c
            @ ^ [X2: c] : ( if_c @ ( member_c @ X2 @ Omega ) @ ( F @ X2 ) @ C2 )
            @ ( sigma_measurable_c_c @ M @ N ) ) ) ) ) ).

% measurable_restrict_space_iff
thf(fact_1141_measurable__restrict__space__iff,axiom,
    ! [Omega: set_c,M: sigma_measure_c,C2: b,N: sigma_measure_b,F: c > b] :
      ( ( member_set_c @ ( inf_inf_set_c @ Omega @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) )
     => ( ( member_b @ C2 @ ( sigma_space_b @ N ) )
       => ( ( member_c_b @ F @ ( sigma_measurable_c_b @ ( sigma_8692839461743104068pace_c @ M @ Omega ) @ N ) )
          = ( member_c_b
            @ ^ [X2: c] : ( if_b @ ( member_c @ X2 @ Omega ) @ ( F @ X2 ) @ C2 )
            @ ( sigma_measurable_c_b @ M @ N ) ) ) ) ) ).

% measurable_restrict_space_iff
thf(fact_1142_measurable__restrict__space__iff,axiom,
    ! [Omega: set_c,M: sigma_measure_c,C2: d,N: sigma_measure_d,F: c > d] :
      ( ( member_set_c @ ( inf_inf_set_c @ Omega @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) )
     => ( ( member_d @ C2 @ ( sigma_space_d @ N ) )
       => ( ( member_c_d @ F @ ( sigma_measurable_c_d @ ( sigma_8692839461743104068pace_c @ M @ Omega ) @ N ) )
          = ( member_c_d
            @ ^ [X2: c] : ( if_d @ ( member_c @ X2 @ Omega ) @ ( F @ X2 ) @ C2 )
            @ ( sigma_measurable_c_d @ M @ N ) ) ) ) ) ).

% measurable_restrict_space_iff
thf(fact_1143_measurable__restrict__space__iff,axiom,
    ! [Omega: set_b,M: sigma_measure_b,C2: c,N: sigma_measure_c,F: b > c] :
      ( ( member_set_b @ ( inf_inf_set_b @ Omega @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
     => ( ( member_c @ C2 @ ( sigma_space_c @ N ) )
       => ( ( member_b_c @ F @ ( sigma_measurable_b_c @ ( sigma_8692839461743104067pace_b @ M @ Omega ) @ N ) )
          = ( member_b_c
            @ ^ [X2: b] : ( if_c @ ( member_b @ X2 @ Omega ) @ ( F @ X2 ) @ C2 )
            @ ( sigma_measurable_b_c @ M @ N ) ) ) ) ) ).

% measurable_restrict_space_iff
thf(fact_1144_measurable__restrict__space__iff,axiom,
    ! [Omega: set_d,M: sigma_measure_d,C2: c,N: sigma_measure_c,F: d > c] :
      ( ( member_set_d @ ( inf_inf_set_d @ Omega @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) )
     => ( ( member_c @ C2 @ ( sigma_space_c @ N ) )
       => ( ( member_d_c @ F @ ( sigma_measurable_d_c @ ( sigma_8692839461743104069pace_d @ M @ Omega ) @ N ) )
          = ( member_d_c
            @ ^ [X2: d] : ( if_c @ ( member_d @ X2 @ Omega ) @ ( F @ X2 ) @ C2 )
            @ ( sigma_measurable_d_c @ M @ N ) ) ) ) ) ).

% measurable_restrict_space_iff
thf(fact_1145_measurable__restrict__space__iff,axiom,
    ! [Omega: set_b,M: sigma_measure_b,C2: d,N: sigma_measure_d,F: b > d] :
      ( ( member_set_b @ ( inf_inf_set_b @ Omega @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
     => ( ( member_d @ C2 @ ( sigma_space_d @ N ) )
       => ( ( member_b_d @ F @ ( sigma_measurable_b_d @ ( sigma_8692839461743104067pace_b @ M @ Omega ) @ N ) )
          = ( member_b_d
            @ ^ [X2: b] : ( if_d @ ( member_b @ X2 @ Omega ) @ ( F @ X2 ) @ C2 )
            @ ( sigma_measurable_b_d @ M @ N ) ) ) ) ) ).

% measurable_restrict_space_iff
thf(fact_1146_measurable__restrict__space__iff,axiom,
    ! [Omega: set_b,M: sigma_measure_b,C2: b,N: sigma_measure_b,F: b > b] :
      ( ( member_set_b @ ( inf_inf_set_b @ Omega @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
     => ( ( member_b @ C2 @ ( sigma_space_b @ N ) )
       => ( ( member_b_b @ F @ ( sigma_measurable_b_b @ ( sigma_8692839461743104067pace_b @ M @ Omega ) @ N ) )
          = ( member_b_b
            @ ^ [X2: b] : ( if_b @ ( member_b @ X2 @ Omega ) @ ( F @ X2 ) @ C2 )
            @ ( sigma_measurable_b_b @ M @ N ) ) ) ) ) ).

% measurable_restrict_space_iff
thf(fact_1147_measurable__restrict__space__iff,axiom,
    ! [Omega: set_d,M: sigma_measure_d,C2: b,N: sigma_measure_b,F: d > b] :
      ( ( member_set_d @ ( inf_inf_set_d @ Omega @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) )
     => ( ( member_b @ C2 @ ( sigma_space_b @ N ) )
       => ( ( member_d_b @ F @ ( sigma_measurable_d_b @ ( sigma_8692839461743104069pace_d @ M @ Omega ) @ N ) )
          = ( member_d_b
            @ ^ [X2: d] : ( if_b @ ( member_d @ X2 @ Omega ) @ ( F @ X2 ) @ C2 )
            @ ( sigma_measurable_d_b @ M @ N ) ) ) ) ) ).

% measurable_restrict_space_iff
thf(fact_1148_measurable__restrict__space__iff,axiom,
    ! [Omega: set_d,M: sigma_measure_d,C2: d,N: sigma_measure_d,F: d > d] :
      ( ( member_set_d @ ( inf_inf_set_d @ Omega @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) )
     => ( ( member_d @ C2 @ ( sigma_space_d @ N ) )
       => ( ( member_d_d @ F @ ( sigma_measurable_d_d @ ( sigma_8692839461743104069pace_d @ M @ Omega ) @ N ) )
          = ( member_d_d
            @ ^ [X2: d] : ( if_d @ ( member_d @ X2 @ Omega ) @ ( F @ X2 ) @ C2 )
            @ ( sigma_measurable_d_d @ M @ N ) ) ) ) ) ).

% measurable_restrict_space_iff
thf(fact_1149_measurable__restrict__space__iff,axiom,
    ! [Omega: set_c,M: sigma_measure_c,C2: set_d,N: sigma_measure_set_d,F: c > set_d] :
      ( ( member_set_c @ ( inf_inf_set_c @ Omega @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) )
     => ( ( member_set_d @ C2 @ ( sigma_space_set_d @ N ) )
       => ( ( member_c_set_d @ F @ ( sigma_6374063253281460273_set_d @ ( sigma_8692839461743104068pace_c @ M @ Omega ) @ N ) )
          = ( member_c_set_d
            @ ^ [X2: c] : ( if_set_d @ ( member_c @ X2 @ Omega ) @ ( F @ X2 ) @ C2 )
            @ ( sigma_6374063253281460273_set_d @ M @ N ) ) ) ) ) ).

% measurable_restrict_space_iff
thf(fact_1150_restrict__space__eq__vimage__algebra,axiom,
    ! [Omega: set_d,M: sigma_measure_d] :
      ( ( ord_less_eq_set_d @ Omega @ ( sigma_space_d @ M ) )
     => ( ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ M @ Omega ) )
        = ( sigma_sets_d
          @ ( sigma_2879403613907127764ra_d_d @ Omega
            @ ^ [X2: d] : X2
            @ M ) ) ) ) ).

% restrict_space_eq_vimage_algebra
thf(fact_1151_restrict__space__eq__vimage__algebra,axiom,
    ! [Omega: set_set_b,M: sigma_measure_set_b] :
      ( ( ord_le3795704787696855135_set_b @ Omega @ ( sigma_space_set_b @ M ) )
     => ( ( sigma_sets_set_b @ ( sigma_5799942775684512035_set_b @ M @ Omega ) )
        = ( sigma_sets_set_b
          @ ( sigma_3488951009927027028_set_b @ Omega
            @ ^ [X2: set_b] : X2
            @ M ) ) ) ) ).

% restrict_space_eq_vimage_algebra
thf(fact_1152_restrict__space__eq__vimage__algebra,axiom,
    ! [Omega: set_b,M: sigma_measure_b] :
      ( ( ord_less_eq_set_b @ Omega @ ( sigma_space_b @ M ) )
     => ( ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ M @ Omega ) )
        = ( sigma_sets_b
          @ ( sigma_8454314777852479828ra_b_b @ Omega
            @ ^ [X2: b] : X2
            @ M ) ) ) ) ).

% restrict_space_eq_vimage_algebra
thf(fact_1153_restrict__space__eq__vimage__algebra_H,axiom,
    ! [M: sigma_measure_b,Omega: set_b] :
      ( ( sigma_sets_b @ ( sigma_8692839461743104067pace_b @ M @ Omega ) )
      = ( sigma_sets_b
        @ ( sigma_8454314777852479828ra_b_b @ ( inf_inf_set_b @ Omega @ ( sigma_space_b @ M ) )
          @ ^ [X2: b] : X2
          @ M ) ) ) ).

% restrict_space_eq_vimage_algebra'
thf(fact_1154_restrict__space__eq__vimage__algebra_H,axiom,
    ! [M: sigma_measure_d,Omega: set_d] :
      ( ( sigma_sets_d @ ( sigma_8692839461743104069pace_d @ M @ Omega ) )
      = ( sigma_sets_d
        @ ( sigma_2879403613907127764ra_d_d @ ( inf_inf_set_d @ Omega @ ( sigma_space_d @ M ) )
          @ ^ [X2: d] : X2
          @ M ) ) ) ).

% restrict_space_eq_vimage_algebra'
thf(fact_1155_restrict__space__eq__vimage__algebra_H,axiom,
    ! [M: sigma_measure_set_b,Omega: set_set_b] :
      ( ( sigma_sets_set_b @ ( sigma_5799942775684512035_set_b @ M @ Omega ) )
      = ( sigma_sets_set_b
        @ ( sigma_3488951009927027028_set_b @ ( inf_inf_set_set_b @ Omega @ ( sigma_space_set_b @ M ) )
          @ ^ [X2: set_b] : X2
          @ M ) ) ) ).

% restrict_space_eq_vimage_algebra'
thf(fact_1156_emeasure__restrict__space,axiom,
    ! [Omega: set_d,M: sigma_measure_d,A: set_d] :
      ( ( member_set_d @ ( inf_inf_set_d @ Omega @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) )
     => ( ( ord_less_eq_set_d @ A @ Omega )
       => ( ( sigma_emeasure_d @ ( sigma_8692839461743104069pace_d @ M @ Omega ) @ A )
          = ( sigma_emeasure_d @ M @ A ) ) ) ) ).

% emeasure_restrict_space
thf(fact_1157_emeasure__restrict__space,axiom,
    ! [Omega: set_set_b,M: sigma_measure_set_b,A: set_set_b] :
      ( ( member_set_set_b @ ( inf_inf_set_set_b @ Omega @ ( sigma_space_set_b @ M ) ) @ ( sigma_sets_set_b @ M ) )
     => ( ( ord_le3795704787696855135_set_b @ A @ Omega )
       => ( ( sigma_emeasure_set_b @ ( sigma_5799942775684512035_set_b @ M @ Omega ) @ A )
          = ( sigma_emeasure_set_b @ M @ A ) ) ) ) ).

% emeasure_restrict_space
thf(fact_1158_emeasure__restrict__space,axiom,
    ! [Omega: set_b,M: sigma_measure_b,A: set_b] :
      ( ( member_set_b @ ( inf_inf_set_b @ Omega @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
     => ( ( ord_less_eq_set_b @ A @ Omega )
       => ( ( sigma_emeasure_b @ ( sigma_8692839461743104067pace_b @ M @ Omega ) @ A )
          = ( sigma_emeasure_b @ M @ A ) ) ) ) ).

% emeasure_restrict_space
thf(fact_1159_null__sets__restrict__space,axiom,
    ! [Omega: set_d,M: sigma_measure_d,A: set_d] :
      ( ( member_set_d @ Omega @ ( sigma_sets_d @ M ) )
     => ( ( member_set_d @ A @ ( measure_null_sets_d @ ( sigma_8692839461743104069pace_d @ M @ Omega ) ) )
        = ( ( ord_less_eq_set_d @ A @ Omega )
          & ( member_set_d @ A @ ( measure_null_sets_d @ M ) ) ) ) ) ).

% null_sets_restrict_space
thf(fact_1160_null__sets__restrict__space,axiom,
    ! [Omega: set_set_b,M: sigma_measure_set_b,A: set_set_b] :
      ( ( member_set_set_b @ Omega @ ( sigma_sets_set_b @ M ) )
     => ( ( member_set_set_b @ A @ ( measur1516554132335629586_set_b @ ( sigma_5799942775684512035_set_b @ M @ Omega ) ) )
        = ( ( ord_le3795704787696855135_set_b @ A @ Omega )
          & ( member_set_set_b @ A @ ( measur1516554132335629586_set_b @ M ) ) ) ) ) ).

% null_sets_restrict_space
thf(fact_1161_null__sets__restrict__space,axiom,
    ! [Omega: set_b,M: sigma_measure_b,A: set_b] :
      ( ( member_set_b @ Omega @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b @ A @ ( measure_null_sets_b @ ( sigma_8692839461743104067pace_b @ M @ Omega ) ) )
        = ( ( ord_less_eq_set_b @ A @ Omega )
          & ( member_set_b @ A @ ( measure_null_sets_b @ M ) ) ) ) ) ).

% null_sets_restrict_space
thf(fact_1162_measurable__restrict__space3,axiom,
    ! [F: b > d,M: sigma_measure_b,N: sigma_measure_d,A: set_b,B: set_d] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ M @ N ) )
     => ( ( member_b_d @ F
          @ ( pi_b_d @ A
            @ ^ [Uu: b] : B ) )
       => ( member_b_d @ F @ ( sigma_measurable_b_d @ ( sigma_8692839461743104067pace_b @ M @ A ) @ ( sigma_8692839461743104069pace_d @ N @ B ) ) ) ) ) ).

% measurable_restrict_space3
thf(fact_1163_measurable__restrict__space3,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,A: set_b,B: set_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( ( member_b_b @ F
          @ ( pi_b_b @ A
            @ ^ [Uu: b] : B ) )
       => ( member_b_b @ F @ ( sigma_measurable_b_b @ ( sigma_8692839461743104067pace_b @ M @ A ) @ ( sigma_8692839461743104067pace_b @ N @ B ) ) ) ) ) ).

% measurable_restrict_space3
thf(fact_1164_measurable__restrict__space3,axiom,
    ! [F: d > b,M: sigma_measure_d,N: sigma_measure_b,A: set_d,B: set_b] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ M @ N ) )
     => ( ( member_d_b @ F
          @ ( pi_d_b @ A
            @ ^ [Uu: d] : B ) )
       => ( member_d_b @ F @ ( sigma_measurable_d_b @ ( sigma_8692839461743104069pace_d @ M @ A ) @ ( sigma_8692839461743104067pace_b @ N @ B ) ) ) ) ) ).

% measurable_restrict_space3
thf(fact_1165_measurable__restrict__space3,axiom,
    ! [F: d > d,M: sigma_measure_d,N: sigma_measure_d,A: set_d,B: set_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ M @ N ) )
     => ( ( member_d_d @ F
          @ ( pi_d_d @ A
            @ ^ [Uu: d] : B ) )
       => ( member_d_d @ F @ ( sigma_measurable_d_d @ ( sigma_8692839461743104069pace_d @ M @ A ) @ ( sigma_8692839461743104069pace_d @ N @ B ) ) ) ) ) ).

% measurable_restrict_space3
thf(fact_1166_measurable__restrict__mono,axiom,
    ! [F: d > b,M: sigma_measure_d,A: set_d,N: sigma_measure_b,B: set_d] :
      ( ( member_d_b @ F @ ( sigma_measurable_d_b @ ( sigma_8692839461743104069pace_d @ M @ A ) @ N ) )
     => ( ( ord_less_eq_set_d @ B @ A )
       => ( member_d_b @ F @ ( sigma_measurable_d_b @ ( sigma_8692839461743104069pace_d @ M @ B ) @ N ) ) ) ) ).

% measurable_restrict_mono
thf(fact_1167_measurable__restrict__mono,axiom,
    ! [F: d > d,M: sigma_measure_d,A: set_d,N: sigma_measure_d,B: set_d] :
      ( ( member_d_d @ F @ ( sigma_measurable_d_d @ ( sigma_8692839461743104069pace_d @ M @ A ) @ N ) )
     => ( ( ord_less_eq_set_d @ B @ A )
       => ( member_d_d @ F @ ( sigma_measurable_d_d @ ( sigma_8692839461743104069pace_d @ M @ B ) @ N ) ) ) ) ).

% measurable_restrict_mono
thf(fact_1168_measurable__restrict__mono,axiom,
    ! [F: b > d,M: sigma_measure_b,A: set_b,N: sigma_measure_d,B: set_b] :
      ( ( member_b_d @ F @ ( sigma_measurable_b_d @ ( sigma_8692839461743104067pace_b @ M @ A ) @ N ) )
     => ( ( ord_less_eq_set_b @ B @ A )
       => ( member_b_d @ F @ ( sigma_measurable_b_d @ ( sigma_8692839461743104067pace_b @ M @ B ) @ N ) ) ) ) ).

% measurable_restrict_mono
thf(fact_1169_measurable__restrict__mono,axiom,
    ! [F: b > b,M: sigma_measure_b,A: set_b,N: sigma_measure_b,B: set_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ ( sigma_8692839461743104067pace_b @ M @ A ) @ N ) )
     => ( ( ord_less_eq_set_b @ B @ A )
       => ( member_b_b @ F @ ( sigma_measurable_b_b @ ( sigma_8692839461743104067pace_b @ M @ B ) @ N ) ) ) ) ).

% measurable_restrict_mono
thf(fact_1170_null__sets__subset,axiom,
    ! [B: set_d,M: sigma_measure_d,A: set_d] :
      ( ( member_set_d @ B @ ( measure_null_sets_d @ M ) )
     => ( ( member_set_d @ A @ ( sigma_sets_d @ M ) )
       => ( ( ord_less_eq_set_d @ A @ B )
         => ( member_set_d @ A @ ( measure_null_sets_d @ M ) ) ) ) ) ).

% null_sets_subset
thf(fact_1171_null__sets__subset,axiom,
    ! [B: set_set_b,M: sigma_measure_set_b,A: set_set_b] :
      ( ( member_set_set_b @ B @ ( measur1516554132335629586_set_b @ M ) )
     => ( ( member_set_set_b @ A @ ( sigma_sets_set_b @ M ) )
       => ( ( ord_le3795704787696855135_set_b @ A @ B )
         => ( member_set_set_b @ A @ ( measur1516554132335629586_set_b @ M ) ) ) ) ) ).

% null_sets_subset
thf(fact_1172_null__sets__subset,axiom,
    ! [B: set_b,M: sigma_measure_b,A: set_b] :
      ( ( member_set_b @ B @ ( measure_null_sets_b @ M ) )
     => ( ( member_set_b @ A @ ( sigma_sets_b @ M ) )
       => ( ( ord_less_eq_set_b @ A @ B )
         => ( member_set_b @ A @ ( measure_null_sets_b @ M ) ) ) ) ) ).

% null_sets_subset
thf(fact_1173_vimage__algebra__sigma,axiom,
    ! [X4: set_set_d,Omega2: set_d,F: b > d,Omega: set_b] :
      ( ( ord_le3937772867789547105_set_d @ X4 @ ( pow_d @ Omega2 ) )
     => ( ( member_b_d @ F
          @ ( pi_b_d @ Omega
            @ ^ [Uu: b] : Omega2 ) )
       => ( ( sigma_8454314777852479830ra_b_d @ Omega @ F
            @ ( sigma_measure_of_d @ Omega2 @ X4
              @ ^ [X2: set_d] : zero_z7100319975126383169nnreal ) )
          = ( sigma_measure_of_b @ Omega
            @ ( collect_set_b
              @ ^ [Uu: set_b] :
                ? [A3: set_d] :
                  ( ( Uu
                    = ( inf_inf_set_b @ ( vimage_b_d @ F @ A3 ) @ Omega ) )
                  & ( member_set_d @ A3 @ X4 ) ) )
            @ ^ [X2: set_b] : zero_z7100319975126383169nnreal ) ) ) ) ).

% vimage_algebra_sigma
thf(fact_1174_vimage__algebra__sigma,axiom,
    ! [X4: set_set_d,Omega2: set_d,F: d > d,Omega: set_d] :
      ( ( ord_le3937772867789547105_set_d @ X4 @ ( pow_d @ Omega2 ) )
     => ( ( member_d_d @ F
          @ ( pi_d_d @ Omega
            @ ^ [Uu: d] : Omega2 ) )
       => ( ( sigma_2879403613907127764ra_d_d @ Omega @ F
            @ ( sigma_measure_of_d @ Omega2 @ X4
              @ ^ [X2: set_d] : zero_z7100319975126383169nnreal ) )
          = ( sigma_measure_of_d @ Omega
            @ ( collect_set_d
              @ ^ [Uu: set_d] :
                ? [A3: set_d] :
                  ( ( Uu
                    = ( inf_inf_set_d @ ( vimage_d_d @ F @ A3 ) @ Omega ) )
                  & ( member_set_d @ A3 @ X4 ) ) )
            @ ^ [X2: set_d] : zero_z7100319975126383169nnreal ) ) ) ) ).

% vimage_algebra_sigma
thf(fact_1175_vimage__algebra__sigma,axiom,
    ! [X4: set_set_d,Omega2: set_d,F: set_b > d,Omega: set_set_b] :
      ( ( ord_le3937772867789547105_set_d @ X4 @ ( pow_d @ Omega2 ) )
     => ( ( member_set_b_d @ F
          @ ( pi_set_b_d @ Omega
            @ ^ [Uu: set_b] : Omega2 ) )
       => ( ( sigma_6754043411966162038et_b_d @ Omega @ F
            @ ( sigma_measure_of_d @ Omega2 @ X4
              @ ^ [X2: set_d] : zero_z7100319975126383169nnreal ) )
          = ( sigma_1409593684724707960_set_b @ Omega
            @ ( collect_set_set_b
              @ ^ [Uu: set_set_b] :
                ? [A3: set_d] :
                  ( ( Uu
                    = ( inf_inf_set_set_b @ ( vimage_set_b_d @ F @ A3 ) @ Omega ) )
                  & ( member_set_d @ A3 @ X4 ) ) )
            @ ^ [X2: set_set_b] : zero_z7100319975126383169nnreal ) ) ) ) ).

% vimage_algebra_sigma
thf(fact_1176_vimage__algebra__sigma,axiom,
    ! [X4: set_set_b,Omega2: set_b,F: b > b,Omega: set_b] :
      ( ( ord_le3795704787696855135_set_b @ X4 @ ( pow_b @ Omega2 ) )
     => ( ( member_b_b @ F
          @ ( pi_b_b @ Omega
            @ ^ [Uu: b] : Omega2 ) )
       => ( ( sigma_8454314777852479828ra_b_b @ Omega @ F
            @ ( sigma_measure_of_b @ Omega2 @ X4
              @ ^ [X2: set_b] : zero_z7100319975126383169nnreal ) )
          = ( sigma_measure_of_b @ Omega
            @ ( collect_set_b
              @ ^ [Uu: set_b] :
                ? [A3: set_b] :
                  ( ( Uu
                    = ( inf_inf_set_b @ ( vimage_b_b @ F @ A3 ) @ Omega ) )
                  & ( member_set_b @ A3 @ X4 ) ) )
            @ ^ [X2: set_b] : zero_z7100319975126383169nnreal ) ) ) ) ).

% vimage_algebra_sigma
thf(fact_1177_vimage__algebra__sigma,axiom,
    ! [X4: set_set_b,Omega2: set_b,F: d > b,Omega: set_d] :
      ( ( ord_le3795704787696855135_set_b @ X4 @ ( pow_b @ Omega2 ) )
     => ( ( member_d_b @ F
          @ ( pi_d_b @ Omega
            @ ^ [Uu: d] : Omega2 ) )
       => ( ( sigma_2879403613907127762ra_d_b @ Omega @ F
            @ ( sigma_measure_of_b @ Omega2 @ X4
              @ ^ [X2: set_b] : zero_z7100319975126383169nnreal ) )
          = ( sigma_measure_of_d @ Omega
            @ ( collect_set_d
              @ ^ [Uu: set_d] :
                ? [A3: set_b] :
                  ( ( Uu
                    = ( inf_inf_set_d @ ( vimage_d_b @ F @ A3 ) @ Omega ) )
                  & ( member_set_b @ A3 @ X4 ) ) )
            @ ^ [X2: set_d] : zero_z7100319975126383169nnreal ) ) ) ) ).

% vimage_algebra_sigma
thf(fact_1178_vimage__algebra__sigma,axiom,
    ! [X4: set_set_b,Omega2: set_b,F: set_b > b,Omega: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ X4 @ ( pow_b @ Omega2 ) )
     => ( ( member_set_b_b @ F
          @ ( pi_set_b_b @ Omega
            @ ^ [Uu: set_b] : Omega2 ) )
       => ( ( sigma_6754043411966162036et_b_b @ Omega @ F
            @ ( sigma_measure_of_b @ Omega2 @ X4
              @ ^ [X2: set_b] : zero_z7100319975126383169nnreal ) )
          = ( sigma_1409593684724707960_set_b @ Omega
            @ ( collect_set_set_b
              @ ^ [Uu: set_set_b] :
                ? [A3: set_b] :
                  ( ( Uu
                    = ( inf_inf_set_set_b @ ( vimage_set_b_b @ F @ A3 ) @ Omega ) )
                  & ( member_set_b @ A3 @ X4 ) ) )
            @ ^ [X2: set_set_b] : zero_z7100319975126383169nnreal ) ) ) ) ).

% vimage_algebra_sigma
thf(fact_1179_sigma__le__iff,axiom,
    ! [A7: set_set_d,Omega: set_d,X: sigma_measure_d] :
      ( ( ord_le3937772867789547105_set_d @ A7 @ ( pow_d @ Omega ) )
     => ( ( ord_le254669808495466590sure_d
          @ ( sigma_measure_of_d @ Omega @ A7
            @ ^ [X2: set_d] : zero_z7100319975126383169nnreal )
          @ X )
        = ( ( ord_less_eq_set_d @ Omega @ ( sigma_space_d @ X ) )
          & ( ( ( sigma_space_d @ X )
              = Omega )
           => ( ord_le3937772867789547105_set_d @ A7 @ ( sigma_sets_d @ X ) ) ) ) ) ) ).

% sigma_le_iff
thf(fact_1180_sigma__le__iff,axiom,
    ! [A7: set_set_set_b,Omega: set_set_b,X: sigma_measure_set_b] :
      ( ( ord_le3201067847557142847_set_b @ A7 @ ( pow_set_b @ Omega ) )
     => ( ( ord_le5713619651007674940_set_b
          @ ( sigma_1409593684724707960_set_b @ Omega @ A7
            @ ^ [X2: set_set_b] : zero_z7100319975126383169nnreal )
          @ X )
        = ( ( ord_le3795704787696855135_set_b @ Omega @ ( sigma_space_set_b @ X ) )
          & ( ( ( sigma_space_set_b @ X )
              = Omega )
           => ( ord_le3201067847557142847_set_b @ A7 @ ( sigma_sets_set_b @ X ) ) ) ) ) ) ).

% sigma_le_iff
thf(fact_1181_sigma__le__iff,axiom,
    ! [A7: set_set_b,Omega: set_b,X: sigma_measure_b] :
      ( ( ord_le3795704787696855135_set_b @ A7 @ ( pow_b @ Omega ) )
     => ( ( ord_le254669799889008988sure_b
          @ ( sigma_measure_of_b @ Omega @ A7
            @ ^ [X2: set_b] : zero_z7100319975126383169nnreal )
          @ X )
        = ( ( ord_less_eq_set_b @ Omega @ ( sigma_space_b @ X ) )
          & ( ( ( sigma_space_b @ X )
              = Omega )
           => ( ord_le3795704787696855135_set_b @ A7 @ ( sigma_sets_b @ X ) ) ) ) ) ) ).

% sigma_le_iff
thf(fact_1182_vimage__algebra__def,axiom,
    ( sigma_8454314777852479830ra_b_d
    = ( ^ [X5: set_b,F2: b > d,M3: sigma_measure_d] :
          ( sigma_measure_of_b @ X5
          @ ( collect_set_b
            @ ^ [Uu: set_b] :
              ? [A3: set_d] :
                ( ( Uu
                  = ( inf_inf_set_b @ ( vimage_b_d @ F2 @ A3 ) @ X5 ) )
                & ( member_set_d @ A3 @ ( sigma_sets_d @ M3 ) ) ) )
          @ ^ [X2: set_b] : zero_z7100319975126383169nnreal ) ) ) ).

% vimage_algebra_def
thf(fact_1183_vimage__algebra__def,axiom,
    ( sigma_8454314777852479828ra_b_b
    = ( ^ [X5: set_b,F2: b > b,M3: sigma_measure_b] :
          ( sigma_measure_of_b @ X5
          @ ( collect_set_b
            @ ^ [Uu: set_b] :
              ? [A3: set_b] :
                ( ( Uu
                  = ( inf_inf_set_b @ ( vimage_b_b @ F2 @ A3 ) @ X5 ) )
                & ( member_set_b @ A3 @ ( sigma_sets_b @ M3 ) ) ) )
          @ ^ [X2: set_b] : zero_z7100319975126383169nnreal ) ) ) ).

% vimage_algebra_def
thf(fact_1184_vimage__algebra__def,axiom,
    ( sigma_2879403613907127764ra_d_d
    = ( ^ [X5: set_d,F2: d > d,M3: sigma_measure_d] :
          ( sigma_measure_of_d @ X5
          @ ( collect_set_d
            @ ^ [Uu: set_d] :
              ? [A3: set_d] :
                ( ( Uu
                  = ( inf_inf_set_d @ ( vimage_d_d @ F2 @ A3 ) @ X5 ) )
                & ( member_set_d @ A3 @ ( sigma_sets_d @ M3 ) ) ) )
          @ ^ [X2: set_d] : zero_z7100319975126383169nnreal ) ) ) ).

% vimage_algebra_def
thf(fact_1185_vimage__algebra__def,axiom,
    ( sigma_2879403613907127762ra_d_b
    = ( ^ [X5: set_d,F2: d > b,M3: sigma_measure_b] :
          ( sigma_measure_of_d @ X5
          @ ( collect_set_d
            @ ^ [Uu: set_d] :
              ? [A3: set_b] :
                ( ( Uu
                  = ( inf_inf_set_d @ ( vimage_d_b @ F2 @ A3 ) @ X5 ) )
                & ( member_set_b @ A3 @ ( sigma_sets_b @ M3 ) ) ) )
          @ ^ [X2: set_d] : zero_z7100319975126383169nnreal ) ) ) ).

% vimage_algebra_def
thf(fact_1186_vimage__algebra__def,axiom,
    ( sigma_6754043411966162038et_b_d
    = ( ^ [X5: set_set_b,F2: set_b > d,M3: sigma_measure_d] :
          ( sigma_1409593684724707960_set_b @ X5
          @ ( collect_set_set_b
            @ ^ [Uu: set_set_b] :
              ? [A3: set_d] :
                ( ( Uu
                  = ( inf_inf_set_set_b @ ( vimage_set_b_d @ F2 @ A3 ) @ X5 ) )
                & ( member_set_d @ A3 @ ( sigma_sets_d @ M3 ) ) ) )
          @ ^ [X2: set_set_b] : zero_z7100319975126383169nnreal ) ) ) ).

% vimage_algebra_def
thf(fact_1187_vimage__algebra__def,axiom,
    ( sigma_6754043411966162036et_b_b
    = ( ^ [X5: set_set_b,F2: set_b > b,M3: sigma_measure_b] :
          ( sigma_1409593684724707960_set_b @ X5
          @ ( collect_set_set_b
            @ ^ [Uu: set_set_b] :
              ? [A3: set_b] :
                ( ( Uu
                  = ( inf_inf_set_set_b @ ( vimage_set_b_b @ F2 @ A3 ) @ X5 ) )
                & ( member_set_b @ A3 @ ( sigma_sets_b @ M3 ) ) ) )
          @ ^ [X2: set_set_b] : zero_z7100319975126383169nnreal ) ) ) ).

% vimage_algebra_def
thf(fact_1188_null__setsI,axiom,
    ! [M: sigma_measure_d,A: set_d] :
      ( ( ( sigma_emeasure_d @ M @ A )
        = zero_z7100319975126383169nnreal )
     => ( ( member_set_d @ A @ ( sigma_sets_d @ M ) )
       => ( member_set_d @ A @ ( measure_null_sets_d @ M ) ) ) ) ).

% null_setsI
thf(fact_1189_null__setsI,axiom,
    ! [M: sigma_measure_b,A: set_b] :
      ( ( ( sigma_emeasure_b @ M @ A )
        = zero_z7100319975126383169nnreal )
     => ( ( member_set_b @ A @ ( sigma_sets_b @ M ) )
       => ( member_set_b @ A @ ( measure_null_sets_b @ M ) ) ) ) ).

% null_setsI
thf(fact_1190_emeasure__neq__0__sets,axiom,
    ! [M: sigma_measure_d,A: set_d] :
      ( ( ( sigma_emeasure_d @ M @ A )
       != zero_z7100319975126383169nnreal )
     => ( member_set_d @ A @ ( sigma_sets_d @ M ) ) ) ).

% emeasure_neq_0_sets
thf(fact_1191_emeasure__neq__0__sets,axiom,
    ! [M: sigma_measure_b,A: set_b] :
      ( ( ( sigma_emeasure_b @ M @ A )
       != zero_z7100319975126383169nnreal )
     => ( member_set_b @ A @ ( sigma_sets_b @ M ) ) ) ).

% emeasure_neq_0_sets
thf(fact_1192_emeasure__notin__sets,axiom,
    ! [A: set_d,M: sigma_measure_d] :
      ( ~ ( member_set_d @ A @ ( sigma_sets_d @ M ) )
     => ( ( sigma_emeasure_d @ M @ A )
        = zero_z7100319975126383169nnreal ) ) ).

% emeasure_notin_sets
thf(fact_1193_emeasure__notin__sets,axiom,
    ! [A: set_b,M: sigma_measure_b] :
      ( ~ ( member_set_b @ A @ ( sigma_sets_b @ M ) )
     => ( ( sigma_emeasure_b @ M @ A )
        = zero_z7100319975126383169nnreal ) ) ).

% emeasure_notin_sets
thf(fact_1194_null__setsD1,axiom,
    ! [A: set_d,M: sigma_measure_d] :
      ( ( member_set_d @ A @ ( measure_null_sets_d @ M ) )
     => ( ( sigma_emeasure_d @ M @ A )
        = zero_z7100319975126383169nnreal ) ) ).

% null_setsD1
thf(fact_1195_null__setsD1,axiom,
    ! [A: set_b,M: sigma_measure_b] :
      ( ( member_set_b @ A @ ( measure_null_sets_b @ M ) )
     => ( ( sigma_emeasure_b @ M @ A )
        = zero_z7100319975126383169nnreal ) ) ).

% null_setsD1
thf(fact_1196_emeasure__eq__0,axiom,
    ! [N: set_d,M: sigma_measure_d,K2: set_d] :
      ( ( member_set_d @ N @ ( sigma_sets_d @ M ) )
     => ( ( ( sigma_emeasure_d @ M @ N )
          = zero_z7100319975126383169nnreal )
       => ( ( ord_less_eq_set_d @ K2 @ N )
         => ( ( sigma_emeasure_d @ M @ K2 )
            = zero_z7100319975126383169nnreal ) ) ) ) ).

% emeasure_eq_0
thf(fact_1197_emeasure__eq__0,axiom,
    ! [N: set_set_b,M: sigma_measure_set_b,K2: set_set_b] :
      ( ( member_set_set_b @ N @ ( sigma_sets_set_b @ M ) )
     => ( ( ( sigma_emeasure_set_b @ M @ N )
          = zero_z7100319975126383169nnreal )
       => ( ( ord_le3795704787696855135_set_b @ K2 @ N )
         => ( ( sigma_emeasure_set_b @ M @ K2 )
            = zero_z7100319975126383169nnreal ) ) ) ) ).

% emeasure_eq_0
thf(fact_1198_emeasure__eq__0,axiom,
    ! [N: set_b,M: sigma_measure_b,K2: set_b] :
      ( ( member_set_b @ N @ ( sigma_sets_b @ M ) )
     => ( ( ( sigma_emeasure_b @ M @ N )
          = zero_z7100319975126383169nnreal )
       => ( ( ord_less_eq_set_b @ K2 @ N )
         => ( ( sigma_emeasure_b @ M @ K2 )
            = zero_z7100319975126383169nnreal ) ) ) ) ).

% emeasure_eq_0
thf(fact_1199_null__sets__def,axiom,
    ( measure_null_sets_d
    = ( ^ [M3: sigma_measure_d] :
          ( collect_set_d
          @ ^ [N3: set_d] :
              ( ( member_set_d @ N3 @ ( sigma_sets_d @ M3 ) )
              & ( ( sigma_emeasure_d @ M3 @ N3 )
                = zero_z7100319975126383169nnreal ) ) ) ) ) ).

% null_sets_def
thf(fact_1200_null__sets__def,axiom,
    ( measure_null_sets_b
    = ( ^ [M3: sigma_measure_b] :
          ( collect_set_b
          @ ^ [N3: set_b] :
              ( ( member_set_b @ N3 @ ( sigma_sets_b @ M3 ) )
              & ( ( sigma_emeasure_b @ M3 @ N3 )
                = zero_z7100319975126383169nnreal ) ) ) ) ) ).

% null_sets_def
thf(fact_1201_sigma__le__sets,axiom,
    ! [A: set_set_d,X4: set_d,N: sigma_measure_d] :
      ( ( ord_le3937772867789547105_set_d @ A @ ( pow_d @ X4 ) )
     => ( ( ord_le3937772867789547105_set_d
          @ ( sigma_sets_d
            @ ( sigma_measure_of_d @ X4 @ A
              @ ^ [X2: set_d] : zero_z7100319975126383169nnreal ) )
          @ ( sigma_sets_d @ N ) )
        = ( ( member_set_d @ X4 @ ( sigma_sets_d @ N ) )
          & ( ord_le3937772867789547105_set_d @ A @ ( sigma_sets_d @ N ) ) ) ) ) ).

% sigma_le_sets
thf(fact_1202_sigma__le__sets,axiom,
    ! [A: set_set_b,X4: set_b,N: sigma_measure_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( pow_b @ X4 ) )
     => ( ( ord_le3795704787696855135_set_b
          @ ( sigma_sets_b
            @ ( sigma_measure_of_b @ X4 @ A
              @ ^ [X2: set_b] : zero_z7100319975126383169nnreal ) )
          @ ( sigma_sets_b @ N ) )
        = ( ( member_set_b @ X4 @ ( sigma_sets_b @ N ) )
          & ( ord_le3795704787696855135_set_b @ A @ ( sigma_sets_b @ N ) ) ) ) ) ).

% sigma_le_sets
thf(fact_1203_sigma__eqI,axiom,
    ! [M: set_set_b,Omega: set_b,N: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ M @ ( pow_b @ Omega ) )
     => ( ( ord_le3795704787696855135_set_b @ N @ ( pow_b @ Omega ) )
       => ( ( ( sigma_sigma_sets_b @ Omega @ M )
            = ( sigma_sigma_sets_b @ Omega @ N ) )
         => ( ( sigma_measure_of_b @ Omega @ M
              @ ^ [X2: set_b] : zero_z7100319975126383169nnreal )
            = ( sigma_measure_of_b @ Omega @ N
              @ ^ [X2: set_b] : zero_z7100319975126383169nnreal ) ) ) ) ) ).

% sigma_eqI
thf(fact_1204_completion__into__space,axiom,
    ! [M: sigma_measure_d] :
      ( ord_le3937772867789547105_set_d
      @ ( collect_set_d
        @ ^ [Uu: set_d] :
          ? [S2: set_d,N3: set_d,N4: set_d] :
            ( ( Uu
              = ( sup_sup_set_d @ S2 @ N3 ) )
            & ( member_set_d @ S2 @ ( sigma_sets_d @ M ) )
            & ( member_set_d @ N4 @ ( measure_null_sets_d @ M ) )
            & ( ord_less_eq_set_d @ N3 @ N4 ) ) )
      @ ( pow_d @ ( sigma_space_d @ M ) ) ) ).

% completion_into_space
thf(fact_1205_completion__into__space,axiom,
    ! [M: sigma_measure_set_b] :
      ( ord_le3201067847557142847_set_b
      @ ( collect_set_set_b
        @ ^ [Uu: set_set_b] :
          ? [S2: set_set_b,N3: set_set_b,N4: set_set_b] :
            ( ( Uu
              = ( sup_sup_set_set_b @ S2 @ N3 ) )
            & ( member_set_set_b @ S2 @ ( sigma_sets_set_b @ M ) )
            & ( member_set_set_b @ N4 @ ( measur1516554132335629586_set_b @ M ) )
            & ( ord_le3795704787696855135_set_b @ N3 @ N4 ) ) )
      @ ( pow_set_b @ ( sigma_space_set_b @ M ) ) ) ).

% completion_into_space
thf(fact_1206_completion__into__space,axiom,
    ! [M: sigma_measure_b] :
      ( ord_le3795704787696855135_set_b
      @ ( collect_set_b
        @ ^ [Uu: set_b] :
          ? [S2: set_b,N3: set_b,N4: set_b] :
            ( ( Uu
              = ( sup_sup_set_b @ S2 @ N3 ) )
            & ( member_set_b @ S2 @ ( sigma_sets_b @ M ) )
            & ( member_set_b @ N4 @ ( measure_null_sets_b @ M ) )
            & ( ord_less_eq_set_b @ N3 @ N4 ) ) )
      @ ( pow_b @ ( sigma_space_b @ M ) ) ) ).

% completion_into_space
thf(fact_1207_null__part__sets_I2_J,axiom,
    ! [S: set_d,M: sigma_measure_d] :
      ( ( member_set_d @ S @ ( sigma_sets_d @ M ) )
     => ( ( sigma_emeasure_d @ M @ ( complete_null_part_d @ M @ S ) )
        = zero_z7100319975126383169nnreal ) ) ).

% null_part_sets(2)
thf(fact_1208_null__part__sets_I2_J,axiom,
    ! [S: set_b,M: sigma_measure_b] :
      ( ( member_set_b @ S @ ( sigma_sets_b @ M ) )
     => ( ( sigma_emeasure_b @ M @ ( complete_null_part_b @ M @ S ) )
        = zero_z7100319975126383169nnreal ) ) ).

% null_part_sets(2)
thf(fact_1209_outer__measure__of__attain,axiom,
    ! [A: set_d,M: sigma_measure_d] :
      ( ( ord_less_eq_set_d @ A @ ( sigma_space_d @ M ) )
     => ? [X3: set_d] :
          ( ( member_set_d @ X3 @ ( sigma_sets_d @ M ) )
          & ( ord_less_eq_set_d @ A @ X3 )
          & ( ( comple3326625653960060653e_of_d @ M @ A )
            = ( sigma_emeasure_d @ M @ X3 ) ) ) ) ).

% outer_measure_of_attain
thf(fact_1210_outer__measure__of__attain,axiom,
    ! [A: set_set_b,M: sigma_measure_set_b] :
      ( ( ord_le3795704787696855135_set_b @ A @ ( sigma_space_set_b @ M ) )
     => ? [X3: set_set_b] :
          ( ( member_set_set_b @ X3 @ ( sigma_sets_set_b @ M ) )
          & ( ord_le3795704787696855135_set_b @ A @ X3 )
          & ( ( comple3418655172853872587_set_b @ M @ A )
            = ( sigma_emeasure_set_b @ M @ X3 ) ) ) ) ).

% outer_measure_of_attain
thf(fact_1211_outer__measure__of__attain,axiom,
    ! [A: set_b,M: sigma_measure_b] :
      ( ( ord_less_eq_set_b @ A @ ( sigma_space_b @ M ) )
     => ? [X3: set_b] :
          ( ( member_set_b @ X3 @ ( sigma_sets_b @ M ) )
          & ( ord_less_eq_set_b @ A @ X3 )
          & ( ( comple3326625653960060651e_of_b @ M @ A )
            = ( sigma_emeasure_b @ M @ X3 ) ) ) ) ).

% outer_measure_of_attain
thf(fact_1212_Un__iff,axiom,
    ! [C2: set_d,A: set_set_d,B: set_set_d] :
      ( ( member_set_d @ C2 @ ( sup_sup_set_set_d @ A @ B ) )
      = ( ( member_set_d @ C2 @ A )
        | ( member_set_d @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_1213_Un__iff,axiom,
    ! [C2: c,A: set_c,B: set_c] :
      ( ( member_c @ C2 @ ( sup_sup_set_c @ A @ B ) )
      = ( ( member_c @ C2 @ A )
        | ( member_c @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_1214_Un__iff,axiom,
    ! [C2: b > d,A: set_b_d,B: set_b_d] :
      ( ( member_b_d @ C2 @ ( sup_sup_set_b_d @ A @ B ) )
      = ( ( member_b_d @ C2 @ A )
        | ( member_b_d @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_1215_Un__iff,axiom,
    ! [C2: set_b,A: set_set_b,B: set_set_b] :
      ( ( member_set_b @ C2 @ ( sup_sup_set_set_b @ A @ B ) )
      = ( ( member_set_b @ C2 @ A )
        | ( member_set_b @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_1216_Un__iff,axiom,
    ! [C2: b > b,A: set_b_b,B: set_b_b] :
      ( ( member_b_b @ C2 @ ( sup_sup_set_b_b @ A @ B ) )
      = ( ( member_b_b @ C2 @ A )
        | ( member_b_b @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_1217_Un__iff,axiom,
    ! [C2: d > b,A: set_d_b,B: set_d_b] :
      ( ( member_d_b @ C2 @ ( sup_sup_set_d_b @ A @ B ) )
      = ( ( member_d_b @ C2 @ A )
        | ( member_d_b @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_1218_Un__iff,axiom,
    ! [C2: d > d,A: set_d_d,B: set_d_d] :
      ( ( member_d_d @ C2 @ ( sup_sup_set_d_d @ A @ B ) )
      = ( ( member_d_d @ C2 @ A )
        | ( member_d_d @ C2 @ B ) ) ) ).

% Un_iff
thf(fact_1219_UnCI,axiom,
    ! [C2: set_d,B: set_set_d,A: set_set_d] :
      ( ( ~ ( member_set_d @ C2 @ B )
       => ( member_set_d @ C2 @ A ) )
     => ( member_set_d @ C2 @ ( sup_sup_set_set_d @ A @ B ) ) ) ).

% UnCI
thf(fact_1220_UnCI,axiom,
    ! [C2: c,B: set_c,A: set_c] :
      ( ( ~ ( member_c @ C2 @ B )
       => ( member_c @ C2 @ A ) )
     => ( member_c @ C2 @ ( sup_sup_set_c @ A @ B ) ) ) ).

% UnCI
thf(fact_1221_UnCI,axiom,
    ! [C2: b > d,B: set_b_d,A: set_b_d] :
      ( ( ~ ( member_b_d @ C2 @ B )
       => ( member_b_d @ C2 @ A ) )
     => ( member_b_d @ C2 @ ( sup_sup_set_b_d @ A @ B ) ) ) ).

% UnCI
thf(fact_1222_UnCI,axiom,
    ! [C2: set_b,B: set_set_b,A: set_set_b] :
      ( ( ~ ( member_set_b @ C2 @ B )
       => ( member_set_b @ C2 @ A ) )
     => ( member_set_b @ C2 @ ( sup_sup_set_set_b @ A @ B ) ) ) ).

% UnCI
thf(fact_1223_UnCI,axiom,
    ! [C2: b > b,B: set_b_b,A: set_b_b] :
      ( ( ~ ( member_b_b @ C2 @ B )
       => ( member_b_b @ C2 @ A ) )
     => ( member_b_b @ C2 @ ( sup_sup_set_b_b @ A @ B ) ) ) ).

% UnCI
thf(fact_1224_UnCI,axiom,
    ! [C2: d > b,B: set_d_b,A: set_d_b] :
      ( ( ~ ( member_d_b @ C2 @ B )
       => ( member_d_b @ C2 @ A ) )
     => ( member_d_b @ C2 @ ( sup_sup_set_d_b @ A @ B ) ) ) ).

% UnCI
thf(fact_1225_UnCI,axiom,
    ! [C2: d > d,B: set_d_d,A: set_d_d] :
      ( ( ~ ( member_d_d @ C2 @ B )
       => ( member_d_d @ C2 @ A ) )
     => ( member_d_d @ C2 @ ( sup_sup_set_d_d @ A @ B ) ) ) ).

% UnCI
thf(fact_1226_sup_Obounded__iff,axiom,
    ! [B2: set_set_b,C2: set_set_b,A2: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ ( sup_sup_set_set_b @ B2 @ C2 ) @ A2 )
      = ( ( ord_le3795704787696855135_set_b @ B2 @ A2 )
        & ( ord_le3795704787696855135_set_b @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1227_sup_Obounded__iff,axiom,
    ! [B2: set_b,C2: set_b,A2: set_b] :
      ( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_set_b @ B2 @ A2 )
        & ( ord_less_eq_set_b @ C2 @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1228_le__sup__iff,axiom,
    ! [X: set_set_b,Y: set_set_b,Z: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ ( sup_sup_set_set_b @ X @ Y ) @ Z )
      = ( ( ord_le3795704787696855135_set_b @ X @ Z )
        & ( ord_le3795704787696855135_set_b @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1229_le__sup__iff,axiom,
    ! [X: set_b,Y: set_b,Z: set_b] :
      ( ( ord_less_eq_set_b @ ( sup_sup_set_b @ X @ Y ) @ Z )
      = ( ( ord_less_eq_set_b @ X @ Z )
        & ( ord_less_eq_set_b @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1230_Un__subset__iff,axiom,
    ! [A: set_set_b,B: set_set_b,C: set_set_b] :
      ( ( ord_le3795704787696855135_set_b @ ( sup_sup_set_set_b @ A @ B ) @ C )
      = ( ( ord_le3795704787696855135_set_b @ A @ C )
        & ( ord_le3795704787696855135_set_b @ B @ C ) ) ) ).

% Un_subset_iff
thf(fact_1231_Un__subset__iff,axiom,
    ! [A: set_b,B: set_b,C: set_b] :
      ( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A @ B ) @ C )
      = ( ( ord_less_eq_set_b @ A @ C )
        & ( ord_less_eq_set_b @ B @ C ) ) ) ).

% Un_subset_iff
thf(fact_1232_sup__inf__absorb,axiom,
    ! [X: set_b,Y: set_b] :
      ( ( sup_sup_set_b @ X @ ( inf_inf_set_b @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_1233_sup__inf__absorb,axiom,
    ! [X: set_d,Y: set_d] :
      ( ( sup_sup_set_d @ X @ ( inf_inf_set_d @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_1234_sup__inf__absorb,axiom,
    ! [X: set_set_b,Y: set_set_b] :
      ( ( sup_sup_set_set_b @ X @ ( inf_inf_set_set_b @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_1235_inf__sup__absorb,axiom,
    ! [X: set_b,Y: set_b] :
      ( ( inf_inf_set_b @ X @ ( sup_sup_set_b @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_1236_inf__sup__absorb,axiom,
    ! [X: set_d,Y: set_d] :
      ( ( inf_inf_set_d @ X @ ( sup_sup_set_d @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_1237_inf__sup__absorb,axiom,
    ! [X: set_set_b,Y: set_set_b] :
      ( ( inf_inf_set_set_b @ X @ ( sup_sup_set_set_b @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_1238_sets_OUn,axiom,
    ! [A2: set_d,M: sigma_measure_d,B2: set_d] :
      ( ( member_set_d @ A2 @ ( sigma_sets_d @ M ) )
     => ( ( member_set_d @ B2 @ ( sigma_sets_d @ M ) )
       => ( member_set_d @ ( sup_sup_set_d @ A2 @ B2 ) @ ( sigma_sets_d @ M ) ) ) ) ).

% sets.Un
thf(fact_1239_sets_OUn,axiom,
    ! [A2: set_b,M: sigma_measure_b,B2: set_b] :
      ( ( member_set_b @ A2 @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b @ B2 @ ( sigma_sets_b @ M ) )
       => ( member_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ ( sigma_sets_b @ M ) ) ) ) ).

% sets.Un
thf(fact_1240_Un__Int__eq_I1_J,axiom,
    ! [S: set_b,T2: set_b] :
      ( ( inf_inf_set_b @ ( sup_sup_set_b @ S @ T2 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_1241_Un__Int__eq_I1_J,axiom,
    ! [S: set_d,T2: set_d] :
      ( ( inf_inf_set_d @ ( sup_sup_set_d @ S @ T2 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_1242_Un__Int__eq_I1_J,axiom,
    ! [S: set_set_b,T2: set_set_b] :
      ( ( inf_inf_set_set_b @ ( sup_sup_set_set_b @ S @ T2 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_1243_Un__Int__eq_I2_J,axiom,
    ! [S: set_b,T2: set_b] :
      ( ( inf_inf_set_b @ ( sup_sup_set_b @ S @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_1244_Un__Int__eq_I2_J,axiom,
    ! [S: set_d,T2: set_d] :
      ( ( inf_inf_set_d @ ( sup_sup_set_d @ S @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_1245_Un__Int__eq_I2_J,axiom,
    ! [S: set_set_b,T2: set_set_b] :
      ( ( inf_inf_set_set_b @ ( sup_sup_set_set_b @ S @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_1246_Un__Int__eq_I3_J,axiom,
    ! [S: set_b,T2: set_b] :
      ( ( inf_inf_set_b @ S @ ( sup_sup_set_b @ S @ T2 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_1247_Un__Int__eq_I3_J,axiom,
    ! [S: set_d,T2: set_d] :
      ( ( inf_inf_set_d @ S @ ( sup_sup_set_d @ S @ T2 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_1248_Un__Int__eq_I3_J,axiom,
    ! [S: set_set_b,T2: set_set_b] :
      ( ( inf_inf_set_set_b @ S @ ( sup_sup_set_set_b @ S @ T2 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_1249_Un__Int__eq_I4_J,axiom,
    ! [T2: set_b,S: set_b] :
      ( ( inf_inf_set_b @ T2 @ ( sup_sup_set_b @ S @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_1250_Un__Int__eq_I4_J,axiom,
    ! [T2: set_d,S: set_d] :
      ( ( inf_inf_set_d @ T2 @ ( sup_sup_set_d @ S @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_1251_Un__Int__eq_I4_J,axiom,
    ! [T2: set_set_b,S: set_set_b] :
      ( ( inf_inf_set_set_b @ T2 @ ( sup_sup_set_set_b @ S @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_1252_Int__Un__eq_I1_J,axiom,
    ! [S: set_b,T2: set_b] :
      ( ( sup_sup_set_b @ ( inf_inf_set_b @ S @ T2 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_1253_Int__Un__eq_I1_J,axiom,
    ! [S: set_d,T2: set_d] :
      ( ( sup_sup_set_d @ ( inf_inf_set_d @ S @ T2 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_1254_Int__Un__eq_I1_J,axiom,
    ! [S: set_set_b,T2: set_set_b] :
      ( ( sup_sup_set_set_b @ ( inf_inf_set_set_b @ S @ T2 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_1255_Int__Un__eq_I2_J,axiom,
    ! [S: set_b,T2: set_b] :
      ( ( sup_sup_set_b @ ( inf_inf_set_b @ S @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_1256_Int__Un__eq_I2_J,axiom,
    ! [S: set_d,T2: set_d] :
      ( ( sup_sup_set_d @ ( inf_inf_set_d @ S @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_1257_Int__Un__eq_I2_J,axiom,
    ! [S: set_set_b,T2: set_set_b] :
      ( ( sup_sup_set_set_b @ ( inf_inf_set_set_b @ S @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_1258_Int__Un__eq_I3_J,axiom,
    ! [S: set_b,T2: set_b] :
      ( ( sup_sup_set_b @ S @ ( inf_inf_set_b @ S @ T2 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_1259_Int__Un__eq_I3_J,axiom,
    ! [S: set_d,T2: set_d] :
      ( ( sup_sup_set_d @ S @ ( inf_inf_set_d @ S @ T2 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_1260_Int__Un__eq_I3_J,axiom,
    ! [S: set_set_b,T2: set_set_b] :
      ( ( sup_sup_set_set_b @ S @ ( inf_inf_set_set_b @ S @ T2 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_1261_Int__Un__eq_I4_J,axiom,
    ! [T2: set_b,S: set_b] :
      ( ( sup_sup_set_b @ T2 @ ( inf_inf_set_b @ S @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_1262_Int__Un__eq_I4_J,axiom,
    ! [T2: set_d,S: set_d] :
      ( ( sup_sup_set_d @ T2 @ ( inf_inf_set_d @ S @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_1263_Int__Un__eq_I4_J,axiom,
    ! [T2: set_set_b,S: set_set_b] :
      ( ( sup_sup_set_set_b @ T2 @ ( inf_inf_set_set_b @ S @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_1264_vimage__Un,axiom,
    ! [F: b > d,A: set_d,B: set_d] :
      ( ( vimage_b_d @ F @ ( sup_sup_set_d @ A @ B ) )
      = ( sup_sup_set_b @ ( vimage_b_d @ F @ A ) @ ( vimage_b_d @ F @ B ) ) ) ).

% vimage_Un
thf(fact_1265_vimage__Un,axiom,
    ! [F: b > b,A: set_b,B: set_b] :
      ( ( vimage_b_b @ F @ ( sup_sup_set_b @ A @ B ) )
      = ( sup_sup_set_b @ ( vimage_b_b @ F @ A ) @ ( vimage_b_b @ F @ B ) ) ) ).

% vimage_Un
thf(fact_1266_vimage__Un,axiom,
    ! [F: d > b,A: set_b,B: set_b] :
      ( ( vimage_d_b @ F @ ( sup_sup_set_b @ A @ B ) )
      = ( sup_sup_set_d @ ( vimage_d_b @ F @ A ) @ ( vimage_d_b @ F @ B ) ) ) ).

% vimage_Un
thf(fact_1267_vimage__Un,axiom,
    ! [F: d > d,A: set_d,B: set_d] :
      ( ( vimage_d_d @ F @ ( sup_sup_set_d @ A @ B ) )
      = ( sup_sup_set_d @ ( vimage_d_d @ F @ A ) @ ( vimage_d_d @ F @ B ) ) ) ).

% vimage_Un

% Helper facts (9)
thf(help_If_2_1_If_001tf__b_T,axiom,
    ! [X: b,Y: b] :
      ( ( if_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__b_T,axiom,
    ! [X: b,Y: b] :
      ( ( if_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001tf__c_T,axiom,
    ! [X: c,Y: c] :
      ( ( if_c @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__c_T,axiom,
    ! [X: c,Y: c] :
      ( ( if_c @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001tf__d_T,axiom,
    ! [X: d,Y: d] :
      ( ( if_d @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__d_T,axiom,
    ! [X: d,Y: d] :
      ( ( if_d @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Set__Oset_Itf__d_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_Itf__d_J_T,axiom,
    ! [X: set_d,Y: set_d] :
      ( ( if_set_d @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_Itf__d_J_T,axiom,
    ! [X: set_d,Y: set_d] :
      ( ( if_set_d @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ord_le3795704787696855135_set_b
    @ ( collect_set_b
      @ ^ [Uu: set_b] :
        ? [A3: set_d] :
          ( ( Uu
            = ( inf_inf_set_b @ ( vimage_b_d @ ( x @ i2 ) @ A3 ) @ ( sigma_space_b @ n ) ) )
          & ( member_set_d @ A3 @ ( sigma_sets_d @ ( m @ i2 ) ) ) ) )
    @ ( sigma_sets_b @ n ) ) ).

%------------------------------------------------------------------------------